Boundedness and absoluteness of some dynamical invariants in model theory
Krzysztof KrupiΕski
[email protected]
Instytut Matematyczny, Uniwersytet WrocΕawski
pl. Grunwaldzki 2/4
50-384 WrocΕaw, Poland
,Β
Ludomir Newelski
[email protected]
Instytut Matematyczny, Uniwersytet WrocΕawski
pl. Grunwaldzki 2/4
50-384 WrocΕaw, Poland
Β andΒ
Pierre Simon
[email protected]
University of California, Berkeley
Mathematics Department, Evans Hall
Berkeley, CA, 94720-3840, USA
Abstract.
Let C be a monster model of an arbitrary theory T, let Ξ±Λ be any (possibly infinite) tuple of bounded length of elements of C, and let cΛ be an enumeration of all elements of C (so a tuple of unbounded length). By SΞ±Λβ(C) we denote the compact space of all complete types over C extending tp(Ξ±Λ/β
), and ScΛβ(C) is defined analogously. Then SΞ±Λβ(C) and ScΛβ(C) are naturally Aut(C)-flows (even Aut(C)-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of C), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called Ο-topology) on the choice of the monster model C; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows SΞ±Λβ(C) and ScΛβ(C). We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of C) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute.
Under the assumption that T has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of ScΛβ(C) is bounded. Then we adapt the proof of [3, Theorem 5.7] to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [10]) from the Ellis group of the flow ScΛβ(C) to the Kim-Pillay Galois group GalKPβ(T) is an isomorphism (in particular, T is G-compact).
We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counter-examples) which occur when the flow ScΛβ(C) is replaced by SΞ±Λβ(C).
Key words and phrases:
Group of automorphisms, Ellis group, minimal flow, boundedness, absoluteness
2010 Mathematics Subject Classification:
03C45, 54H20, 37B05
The first and second authors are supported by the Narodowe Centrum Nauki grant no. 2015/19/B/ST1/01151; the first author is also supported by the Narodowe Centrum Nauki grant no. 2016/22/E/ST1/00450
The third author was partially supported by ValCoMo (ANR-13-BS01-0006).
0. Introduction
Some methods and ideas of topological dynamics were introduced to
model theory by the second author in [12, 13, 14]. For a group G definable in a first order structure M the fundamental object in this approach is the G-flow SGβ(M) (the space of complete types over M concentrated on G) or rather SG,extβ(M) (the space of all external types over M concentrated on G). Then the Ellis semigroup E(SG,extβ(M)) of this flow, minimal left ideals of this semigroup, and the Ellis group play the key role in this approach. The motivation for these considerations is the fact that various notions and ideas from topological dynamics lead to new interesting objects and phenomena in model theory. In particular, this allows us to extend some aspects of the theory of stable groups to a much more general context. Another context to apply ideas of topological dynamics to model theory is via looking at the action of the group Aut(C) of automorphisms of a monster model C of a given (arbitrary) theory T on various spaces of types over C. Some aspects of this approach appeared already in [15]. More recently, this approach was developed and applied in [10] to prove very general theorems about spaces of arbitrary strong types (i.e. spaces of classes of bounded invariant equivalence relations refining type), answering in particular some questions from earlier papers. So this is an example where the topological dynamics approach to model theory not only leads to a development of a new theory, but can also be used to solve existing problems in model theory which were not accessible by previously known methods.
From the model-theoretic perspective, an important kind of question
(raised and considered by the second author and later also by
Chernikov and the third author in [3]) is to what extent various dynamical invariants have a model-theoretic nature, e.g. in the case of a group G definable in M, what are the relationships between the minimal left ideals of the Ellis semigroups, or between the Ellis groups, of the flow SG,extβ(M) when the ground model M varies: are some of these objects invariant under changing M, do there exist epimorphisms from objects computed for a bigger model onto the corresponding objects computed for a smaller model? This turned out to be difficult to settle in general and there are only some partial results in [13, 14]; a quite comprehensive understanding of these issues appears in [3] in the context of definably amenable groups in NIP theories, e.g. in this context the Ellis group does not depend on the choice of the ground model M and is isomorphic to the so-called definable Bohr compactification of G.
In this paper, we study these kind of questions for the second context mentioned above, namely for the group Aut(C) acting on some spaces of complete types over C, and, in contrast with the case of a definable group G, here we answer most of these questions in full generality. Some of these questions were formulated at the end of Section 2 of [10].
In the rest of this introduction, we will state precisely in which questions we are interested in and what our main results are. For an exposition of basic notions and facts on topological dynamics which are essential in this paper the reader is referred to [10, Subsection 1.1].
Let C be a monster model of a theory T, and let cΛ be an enumeration of all elements of C. Recall that a monster model is a ΞΊ-saturated and strongly ΞΊ-homogeneous model for a βbig enoughβ strong limit cardinal ΞΊ. A cardinal is said to be bounded (with respect to C) if it is less than ΞΊ.
Then ScΛβ(C):={tp(aΛ/C):aΛβ‘cΛ} is an Aut(C)-ambit. By EL we denote the Ellis semigroup of this flow. Let M be a minimal left ideal of EL and uβM an idempotent; so uM is the Ellis group of the flow ScΛβ(C). This group played a fundamental role in [10], where a continuous epimorphism f:uMβGalLβ(T) (where GalLβ(T) is the Lascar Galois group of T) was found and used to understand the descriptive set-theoretic complexity of GalLβ(T) and of spaces of arbitrary strong types.
The group uM can be equipped with the so-called Ο-topology, and then uM/H(uM) is a compact (Hausdorff) topological group, where H(uM) is the intersection of the Ο-closures of the Ο-neighborhoods of u. In fact, an essential point in [10] was that f factors through H(uM).
We will say that an object or a property defined in terms of C is absolute if it does not depend (up to isomorphism, if it makes sense) on the choice of C. One can think that absoluteness means that an object or a property is a model-theoretic invariant of the theory in question.
In Section 4, we study Ellis groups.
Question 0.1**.**
*i) Does the Ellis group uM of the flow ScΛβ(C) have bounded size?
ii) Is this Ellis group independent of the choice of C as a group equipped with the Ο-topology?
iii) Is the compact topological group uM/H(uM) independent (as a topological group) of the choice of C?*
The positive answer to (ii) clearly implies that the answer to (iii) is also positive.
One can formulate the same questions also with cΛ replaced by a
tuple Ξ±Λ of elements of C, of bounded length (we call
such a tuple Ξ± short). We will prove that the answers to all these questions are positive (with no assumptions on the theory T). In fact, we will deduce it from a more general theorem. Namely, consider any product S of
sorts Siβ, iβI (possibly with unboundedly many factors, with repetitions allowed), and let xΛ=(xiβ)iβIβ be the corresponding tuple of variables (i.e. xiβ is from the sort Siβ). By SSβ(C) or SxΛβ(C) we denote the space of all complete types over C of tuples from S. Then SSβ(C) is an Aut(C)-flow.
In this paper, by a β
-type-definable subset X of S we will mean a partial β
-type in the variables xΛ, and when S has only boundedly many factors, X will be freely identified with the actual subset X(C) of S computed in C. The space SXβ(C) of complete types over C concentrated on X is an Aut(C)-subflow of the flow SSβ(C). Of course, it makes sense to consider SXβ(Cβ²) for an arbitrary monster model Cβ². One of our main results is
Theorem 0.2**.**
Let X be any β
-type-definable subset of S. Then the Ellis group of the Aut(C)-flow SXβ(C) is of bounded size and does not depend (as a group equipped with the Ο-topology) on the choice of the monster model C.
Moreover, we will see that
βΆ5β(β£Tβ£) is an absolute bound on the size of this Ellis group. (Where βΆ0β(ΞΊ)=ΞΊ and βΆn+1β(ΞΊ)=2βΆnβ(ΞΊ).)
From this theorem, we will easily deduce that the answer to Question 0.1, and also to its counterpart for cΛ replaced by a short tuple Ξ±Λ, is positive, and, moreover, that
βΆ5β(β£Tβ£) is an absolute bound on the size of the Ellis group in both these cases.
Note that there is no counterpart of this behavior in the case of a
group G definable in M, namely it may happen that the sizes of the
Ellis groups of the flows SG,extβ(M) are getting arbitrarily
large when M varies. For example, it is the case whenever the
component Gβ00 does not exist. Indeed, in this case the groups
Gβ/GMββ00 are getting arbitrarily large when M grows, and
by [12] they are homomorphic images of the Ellis groups of SG,extβ(M).
The fundamental tool in the proof Theorem 0.2 is the notion of content of a sequence of types introduced in Section 3. This gives a model-theoretic characterization of when a type is in the orbit closure of another one. It also allows us to define an independence relation which makes sense in any theory. The properties of this relation will be studied in a future work.
In Section 5, we focus on minimal left ideals in Ellis semigroups. As above, let M be a minimal left ideal of the Ellis semigroup EL of the flow ScΛβ(C). Recall that all minimal left ideals of EL are isomorphic as Aut(C)-flows (so they are of the same size), but they need not be isomorphic as semigroups.
Question 0.3**.**
*i) Is the size of M bounded?
ii) Is the property that M is of bounded size absolute?
iii) If M is of bounded size and the answer to (ii) is positive, what are the relationships between the minimal left ideals when C varies?*
As above, one has the same question for cΛ replaced by any short tuple Ξ±Λ. We give an easy example showing that the answer to (i) is negative (see Example 5.1). Then we answer positively (ii) and give an answer to (iii) for both cΛ and Ξ±Λ. As above, we will deduce these things from the following, more general result.
Theorem 0.4**.**
*Let X be any β
-type-definable subset of S.
i) The property that a minimal left ideal of the Ellis semigroup of the flow SXβ(C) is of bounded size is absolute.
ii) Assume that a minimal left ideal of the Ellis semigroup of SXβ(C) is of bounded size. Let C1β and C2β be two monster models. Then for every minimal left ideal M1β of the Ellis semigroup of the flow SXβ(C1β) there exists a minimal left ideal M2β of the Ellis semigroup of the flow SXβ(C2β) which is isomorphic to M1β as a semigroup.*
Moreover, we show that if a minimal left ideal of the Ellis semigroup of the flow SXβ(C) is of bounded size, then this size is bounded by βΆ3β(β£Tβ£).
The above theorem together with Example 5.1 leads to the following problem which we study mostly in Section 7. (In between, in Section 6, we easily describe the minimal left ideals, the Ellis group, etc. in the case of stable theories.)
Problem 0.5**.**
*i) Characterize when a minimal left ideal of the Ellis semigroup of the flow SXβ(C) is of bounded size.
ii) Do the same for the flow ScΛβ(C).
iii) Do the same for the flow SΞ±Λβ(C), where Ξ±Λ is a short tuple from C.*
We will easily see that that a solution of (i) would yield solutions of (ii) and (iii). As to (i), we observe in Section 5 that such an ideal is of bounded size if and only if there is an element in the Ellis semigroup of SXβ(C) which maps all types from SXβ(C) to Lascar invariant types. Then, in Section 7, under the additional assumption that T has NIP, we give several characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of the flow ScΛβ(C) is of bounded size (i.e. we solve Problem 0.5(ii) for NIP theories). This is done in Theorem 7.7. A part of this theorem is the following.
Proposition 0.6**.**
Assume that T has NIP. Then a minimal left ideal of the Ellis semigroup of the flow ScΛβ(C) is of bounded size if and only if β
is an extension base (i.e. every type over β
does not fork over β
).
We also get some characterizations of boundedness of the minimal left ideals of the Ellis semigroup of SXβ(C), which lead us to natural questions (see Questions 7.5 and 7.10).
In the second subsection of Section 7, we get a better bound on the size of the Ellis group of the flow SXβ(C) under the NIP assumption, namely
βΆ3β(β£Tβ£). In the case of the Ellis groups of the flows ScΛβ(C) and SΞ±Λβ(C), we get a yet smaller bound, namely 22β£Tβ£.
In the last subsection of Section 7, we recall from
[10] a natural epimorphism F from the Ellis group of the
flow ScΛβ(C) to the Kim-Pillay Galois group GalKPβ(T). We adapt the proof of Theorem 5.7 from [3] to show:
Theorem 0.7**.**
Assume that T has NIP. If a minimal left ideal of the Ellis
semigroup of the flow ScΛβ(C) is of bounded size, then the
epimorphism F mentioned above is an isomorphism.
Since the map F is the composition of some natural homomoprhism from the Ellis group of the flow ScΛβ(C) to GalLβ(T) with the natural map GalLβ(T)βGalKPβ(T), as an immediate corollary we get that under the assumtpions of
Theorem 0.7, T is G-compact. Alternatively, G-compactness follows from Proposition 0.6 and [7, Corollary 2.10].
On the other hand, taking any non G-compact theory, although by Theorem 0.2 the Ellis group of the flow ScΛβ(C) is always bounded, we get that F need not be an isomorphism (even in the NIP context, as there exist non G-compact NIP theories). This shows that the assumption that a minimal left ideal is of bounded size is essential in the last theorem. Using some non-trivial results from [10], we can conclude more, namely that the natural epimorphism from the Ellis group onto GalLβ(T) need not be an isomorphism, even in the NIP context.
We also describe an epimorphism from the Ellis group of the flow SΞ±Λβ(C) to the new Kim-Pillay Galois group introduced in [4] and denoted by GalKPfix,1β(p), where p=tp(Ξ±Λ/β
). This is a natural counterpart of the epimorphism considered in Theorem 0.7. However, we give an example showing that the obvious counterpart of Theorem 0.7 does not hold for this new epimorphism, namely assuming that T has NIP and a minimal left ideal of the Ellis semigroup of the flow SΞ±Λβ(C) is of bounded size, the new epimorphism need not be an isomorphism.
1. Some preliminaries
For the relevant notions and facts from topological dynamics the reader is referred to [10, Subsection 1.1]. (We do not include it here, because this would be the exact copy.)
Let us only say that the Ellis semigroup of a flow (G,X) will be
denoted by EL(X), a minimal left ideal in EL(X) will be usually
denoted by M (sometimes with some indexes) and
idempotents in M will be denoted by u or v. We should
emphasize that all minimal left ideals (equivalently, minimal
subflows) of EL(X) are isomorphic as G-flows [6, Proposition
I.2.5] but not necessarily
as semigroups. The fact that they are isomorphic as G-flows implies that they are of the same size, and so, in the whole paper, the statement βa minimal left ideal of a given Ellis semigroup is of bounded sizeβ is equivalent to the statement βthe minimal left ideals of the given Ellis semigroup are of bounded sizeβ.
In this paper, we will often consider nets indexed by some formulas Ο(xΛ,bΛ). Formally, this means that on the set of formulas treated as elements of the Lidenbaum algebra (i.e. equivalent formals are identified) we take the natural directed order: Οβ€Ο if and only if Οβ’Ο. But throughout this paper, we will just talk about formulas, not always mentioning that we are working in the Lidenbaum algebra. The same remark concerns nets indexed by finite sequences of formulas (where the order is formally the product of orders on the Lindenbaum algebras in every coordinate). In fact, we could work just with formulas and with nets indexed by preorders, but we find more elegant to use orders.
Now, we give a few details on Galois groups in model theory. For more
detailed expositions the reader is referred to [10, Subsection
1.3] or [9, Subsection 4.1]. If the reader is
interested in yet more details and proofs, he or she may consult
fundamental papers around this topic, e.g. [11, 17] or [1].
Let C be a monster model of a theory T.
Definition 1.1**.**
- i)
The group of Lascar strong automorphisms, denoted by AutfLβ(C), is the subgroup of Aut(C) generated by all automorphisms fixing a small submodel of C pointwise, i.e.Β AutfLβ(C)=β¨Ο:ΟβAut(C/M)\mboxforasmallMβΊCβ©.
2. ii)
The Lascar Galois group of T, denoted by GalLβ(T), is the quotient group Aut(C)/AutfLβ(C) (which makes sense, as AutfLβ(C) is a normal subgroup of Aut(C)).
The orbit equivalence relation of AutfLβ(C) acting on any given product S of boundedly many sorts of C is usually denoted by ELβ. It turns out that this is the finest bounded (i.e. with boundedly many classes), invariant equivalence relation on S; and the same is true after the restriction to the set of realizations of any type in SSβ(β
). The classes of ELβ are called Lascar strong types.
Now, we recall the logic topology on GalLβ(T).
Let Ξ½:Aut(C)βGalLβ(T) be the quotient map. Choose a small model M, and let mΛ be its enumeration. By SmΛβ(M) we denote {tp(nΛ/M):nΛβ‘mΛ}. Let Ξ½1β:Aut(C)βSmΛβ(M) be defined by Ξ½1β(Ο)=tp(Ο(mΛ)/M), and Ξ½2β:SmΛβ(M)βGalLβ(T) by Ξ½2β(tp(Ο(mΛ)/M))=Ο/AutfLβ(C). Then Ξ½2β is a well-defined surjection, and Ξ½=Ξ½2ββΞ½1β. Thus, GalLβ(T) becomes the quotient of the space SmΛβ(M) by the relation of lying in the same fiber of Ξ½2β, and so we can define a topology on GalLβ(T) as the quotient topology. In this way, GalLβ(T) becomes a quasi-compact (so not necessarily Hausdorff) topological group. This topology does not depend on the choice of the model M.
Next, define Gal0β(T) as the closure of the identity in GalLβ(T). We put AutfKPβ(T):=Ξ½β1[Gal0β(T)], and finally GalKPβ(T):=Aut(C)/AutfKPβ(C). Then GalKPβ(T)β
GalLβ(T)/Gal0β(T) becomes a compact (Hausdorff) topological group (with the quotient topology). We also have the obvious continuous epimorphism h:GalLβ(T)βGalKPβ(T). We say that T is G-compact if h is an isomorphism; equivalently, if Gal0β(T) is trivial.
Taking M of cardinality β£Tβ£, since GalLβ(T) is the image of SmΛβ(M) by Ξ½2β, we get that β£GalLβ(T)β£β€2β£Tβ£.
Finally, we recall some results from [10]. As usual, cΛ is an enumeration of C. Let EL=EL(ScΛβ(C)), M be a minimal left ideal in EL, and uβM an idempotent. Let Cβ²β»C be a monster model with respect to C.
We define f^β:ELβGalLβ(T) by
[TABLE]
where Οβ²βAut(Cβ²) is such that Οβ²(cΛ)β¨Ξ·(tp(cΛ/C)). It turns out that this is a
well-defined semigroup epimorphism. Its restriction f to
uM is a group epimorphism from uM to
GalLβ(T) [10, Proposition 2.4 and Corollary 2.6].
Fact 1.2** (Theorem 2.7 from [10]).**
Equip uM with the Ο-topology and uM/H(uM) with the induced quotient topology. Then:
- (1)
f* is continuous.*
2. (2)
H(uM)β€ker(f).
3. (3)
The formula p/H(uM)β¦f(p) yields a well-defined continuous epimorphism fΛβ from uM/H(uM) to GalLβ(T).
In particular, we get the following sequence of continuous epimorphisms:
[TABLE]
Fact 1.3** (Theorem 2.9 from [10]).**
For Y:=ker(fΛβ) let clΟβ(Y) be the closure of Y inside uM/H(uM). Then fΛβ[clΟβ(Y)]=Gal0β(T), so fΛβ restricted to clΟβ(Y) induces an isomorphism between clΟβ(Y)/Y and Gal0β(T).
These facts imply that GalLβ(T) can be presented as the quotient of a compact Hausdorff group by some subgroup, and Gal0β(T) is such a quotient but by a dense subgroup.
Corollary 1.4**.**
If fΛβ is an isomorphism, then Gal0β(T) is trivial (i.e. T is G-compact). In particular, if f is an isomorphism, then T is G-compact.
Proof.
If fΛβ is an isomorphism, then Y is a singleton. But the Ο-topology on uM is T1β, so Y is Ο-closed, i.e. clΟβ(Y)=Y has only one element. Hence, by Fact 1.3, Gal0β(T)=fΛβ[clΟβ(Y)] is trivial.
β
2. A few reductions
We explain here some basic issues which show that Theorems 0.2 and 0.4 yield answers to Questions 0.1 and 0.3, and that it is enough to prove these theorems assuming that the number of factors in the product S is bounded.
Let C be a monster model of an arbitrary theory T.
Recall that cΛ is an enumeration of C, and Ξ±Λ is a short tuple of elements of C.
Whenever we talk about realizations of types over C, we choose them from a bigger monster model Cβ²β»C.
If p(yΛβ) is a type and xΛβyΛβ is a subsequence of variables, we let pβ£xΛβ denote the restriction of p to the variables xΛ.
The following remark is easy to check.
Remark 2.1*.*
Let dΛ be a tuple of all elements of C (with repetitions) such that Ξ±Λ is a subsequence of dΛ. Let r:SdΛβ(C)βSΞ±Λβ(C) be the restriction to the appropriate coordinates. Then r is an epimorphism of Aut(C)-ambits which induces an epimorphism r^:EL(SdΛβ(C))βEL(SΞ±Λβ(C)) of semigroups. In particular, each minimal left ideal in EL(SdΛβ(C)) maps via r^ onto a minimal left ideal in EL(SΞ±Λβ(C)), and similarly for the Ellis groups.
Corollary 2.2**.**
The size of a minimal left ideal in EL(ScΛβ(C)) is greater than or equal to the size of a minimal left ideal in EL(SΞ±Λβ(C)), and the size of the Ellis group of the flow ScΛβ(C) is greater than or equal to the size of the Ellis group of the flow SΞ±Λβ(C).
Let S,Sβ² be two products of sorts, with possibly an unbounded number of factors and repetitions allowed, with associated variables xΛ and yΛβ respectively. Let X [resp. Y] be a β
-type-definable subset of S [resp. Sβ²]. Say that X and Y have the same finitary content if for every finite xΛβ²βxΛ and pβSXβ(C) there is a yΛββ²βyΛβ of the same size (and associated with the same sorts) and qβSYβ(C) such that pβ£xΛβ²β=qβ£yΛββ²β and conversely, switching the roles of X and Y (formally, this equality denotes equality after the identification of the corresponding variables, but we will usually ignore this). The above notion of finitary content has nothing to do with the notion of content of a type which will be introduced in the next section.
Proposition 2.3**.**
Let S,Sβ², X and Y be as above and assume that X and Y have the same finitary content. Then EL(SXβ(C))β
EL(SYβ(C)) as semigroups and as Aut(C)-flows. In particular, the corresponding minimal left ideals of these Ellis semigroups are isomorphic, and the Ellis groups of the flows SXβ(C) and SYβ(C) are isomorphic as groups equipped with the Ο-topology.
Proof.
Take Ξ·βEL(SXβ(C)), p(xΛ),q(xΛ)βSXβ(C). Let xΛ0β,xΛ1ββxΛ be two finite tuples of the same size. Assume that pβ£xΛ0ββ=qβ£xΛ1ββ. Then Ξ·(p)β£xΛ0ββ=Ξ·(q)β£xΛ1ββ. This allows us to define f:EL(SXβ(C))βEL(SYβ(C)) by putting f(Ξ·)=Ξ·β², where for every q(yΛβ)βSYβ(C) and finite yΛββ²βyΛβ, Ξ·β²(q)β£yΛββ²β=Ξ·(p)β£xΛβ²β, where pβSXβ(C) and xΛβ²βxΛ are such that pβ£xΛβ²β=qβ£yΛββ²β.
Now, f is a morphism of semigroups and of
Aut(C)-flows. Furthermore, the map g:EL(SYβ(C))βEL(SXβ(C)) defined in the same way as f switching the
roles of X and Y, is an inverse of f. Therefore, f is
an isomorphism of semigroups and Aut(C)-flows. Also, f
respects the Ο-topology. This follows immediately from
the definition of the Ο-topology (see
[10, Definitions 1.3 and 1.4]) and the fact that f maps id to
id.
β
Proposition 2.4**.**
Let S be the product of all the sorts of the language such that each sort is repeated β΅0β times. Then EL(ScΛβ(C))β
EL(SSβ(C)) as semigroups and as Aut(C)-flows. In particular, the corresponding minimal left ideals of these Ellis semigroups are isomorphic, and the Ellis groups of the flows ScΛβ(C) and SSβ(C) are isomorphic as groups equipped with the Ο-topology.
Proof.
This follows at once from Proposition 2.3, because EL(ScΛβ(C))β
EL(SdΛβ(C)), and tp(dΛ/β
) and S have the same
finitary content, where dΛ is the tuple of all elements C such
that each element is repeated β΅0β times.
β
Proposition 2.3 also implies that EL(ScΛβ(C)) is isomorphic with EL(SΞ²Λββ(C)) for a suitably chosen short tuple Ξ²Λβ. Namely, we have
Proposition 2.5**.**
Let mΛ be an enumeration of an β΅0β-saturated model (of bounded size). Then EL(ScΛβ(C))β
EL(SmΛβ(C)) as semigroups and as Aut(C)-flows, and we have all the further conclusions as in Proposition 2.4.
To complete the picture, we finish with the following proposition which shows that in most of our results without loss of generality one can assume that the product of sorts in question has only boundedly many factors.
Actually, the next proposition is βalmostβ a generalization of Proposition 2.4. Note, however, that the bound on the number of factors in the obtained product is smaller in Proposition 2.4, and an important information in Proposition 2.4 is that the obtained product of sorts does not depend on the choice of C and its enumeration cΛ (although the product of sorts to which cΛ belongs is getting arbitrarily long when C is getting bigger).
Proposition 2.6**.**
Let S be a product of some sorts of the language with repetitions allowed so that the number of factors may be unbounded, and let X be a β
-type-definable subset of this product. Then there exists a product Sβ² of some sorts with a bounded number (at most 2β£Tβ£) of factors and a β
-type-definable subset Y of Sβ² such that EL(SXβ(C))β
EL(SYβ(C)) as semigroups and as Aut(C)-flows. In particular, the corresponding minimal left ideals of these Ellis semigroups are isomorphic, and the Ellis groups of the flows SXβ(C) and SYβ(C) are isomorphic as groups equipped with the Ο-topology. Moreover, Sβ² and Y can be chosen independently of the choice of C (for the given S and X).
Proof.
Let S=βi<Ξ»βSiβ and let xΛ=(xiβ)i<Ξ»β be the variables of types in SSβ(C). For any two finite subsequences yΛβ and zΛ of xΛ of the same length and associated with the same sorts, any two types q(zΛ) and r(yΛβ) will be identified if and only if they are equal after the identification of zΛ and yΛβ.
For any subsequence yΛβ=(xΛiβ)iβIβ of xΛ, let ΟyΛββ denote the projection from X to βiβIβSiβ. Then ΟyΛββ[X] is a β
-type-definable subset of βiβIβSiβ. Moreover, the restriction map ryΛββ:SXβ(C)βSβiβIβSiββ(C) (i.e. ryΛββ(p):=pβ£yΛββ) maps SXβ(C) onto SΟyΛββ[X]β(C).
Now, there exists IβΞ» (independent of the choice of C) of cardinality at most 2β£Tβ£ such that for any finite sequence of sorts PΛ the collection of all sets ΟzΛβ[X], where zΛ is a finite subsequence of xΛ of variables associated with the sorts PΛ, coincides with the collection of such sets with zΛ ranging over all finite subsequences (of variables associated with the sorts PΛ) of the tuple yΛβ:=(xΛ)iβIβ. Let Y=ΟyΛββ[X] be the projection of X to the product of sorts Sβ²:=βiβIβSiβ. Then Y is a β
-type-definable subset of Sβ². By construction, Y has the same finitary content as X, hence by Proposition 2.3, EL(SXβ(C)) and EL(SYβ(C)) are isomorphic as semigroups and Aut(C)-flows.
β
3. Content of types and subflows of SXβ(C)
In this section, we define the content of a type (or a tuple of types) and use it to understand subflows of the space of types.
As usual, we work in a monster model C of an arbitrary theory T.
Definition 3.1**.**
Let AβB.
- i)
Let p(xΛ)βS(B). Define the content of p over A as
[TABLE]
where Ο(xΛ,yΛβ) are formulas with parameters from A.
When A=β
, we write simply c(p) and call it the content of p.
2. ii)
If the pair (Ο(xΛ,yΛβ),q(yΛβ)) is in cAβ(p), then we say that it is represented in p(xΛ).
3. iii)
Likewise, for a sequence of types p1β(xΛ),β¦,pnβ(xΛ)βS(B), q(yΛβ)βS(A) and a sequence Ο1β(xΛ,yΛβ),β¦,Οnβ(xΛ,yΛβ) of formulas with parameters from A, we say that (Ο1β,β¦,Οnβ,q) is represented in (p1β,β¦,pnβ) if Ο1β(xΛ,bΛ)βp1β,β¦,Οnβ(xΛ,bΛ)βpnβ for some bΛβ¨q. We define the contents cAβ(p1β,β¦,pnβ) and c(p1β,β¦,pnβ) accordingly.
The relation c(p)βc(q) is similar to the fundamental order as defined by Lascar and Poizat ([16]) as an alternative approach to forking in stable theories. More precisely, the relation of inclusion of content is a refinement of the usual fundamental order. With this analogy in mind, we define analogue notions of heirs and coheirs.
Definition 3.2**.**
Let MβA and p(xΛ)βS(A).
- i)
We say that p(xΛ) is a strong heir over M if for every finite mΛβM and Ο(xΛ,aΛ)βp(xΛ), aΛ finite, there is aΛβ²βM with Ο(xΛ,aΛβ²)βp(xΛ) such that tp(aΛ/mΛ)=tp(aΛβ²/mΛ).
2. ii)
We say that p(xΛ)βS(A) is a strong coheir over M if for every finite mΛβM and every Ο(xΛβ²,aΛ)βp(xΛ), where xΛβ²βxΛ is a finite subsequence of variables, there is a bΛβM realizing Ο(xΛβ²,aΛ), with tp(bΛ/mΛ)=pβ£mΛβ(xΛβ²).
In this definition Ο(xΛ,yΛβ) denotes a formula over
β
, but equivalently we may assume that Ο(xΛ,yΛβ)
is over M.
Note that those notions are dual:
[TABLE]
Lemma 3.3**.**
Let MβA, where M is β΅0β-saturated. Assume that p(xΛ)βS(M). Then p(xΛ) has an extension pβ²(xΛ)βS(A) which is a strong coheir [resp. strong heir] over M.
Proof.
Choose an ultrafilter U on Mβ£xΛβ£ containing all sets of the form pβ£mΛβ(xΛβ²)(M), where mΛβM and xΛβ²βxΛ are finite. Define pβ²(xΛ) as the set of formulas Ο(xΛ) with parameters from A such that Ο(C)β©Mβ£xΛβ£βU. Then pβ² extends p and is a strong coheir over M.
To construct a strong heir of p, we dualize the
argument. Let bΛβ¨p(xΛ) and let aΛ
enumerate A. By the previous paragraph, let aΛβ²β‘MβaΛ be such that tp(aΛβ²/MbΛ) is a strong coheir over M. Then tp(bΛ/MaΛβ²) is a strong heir over M. Take fβAut(C/M) mapping aΛβ² to aΛ. Then r(xΛ):=tp(f(bΛ)/MaΛ) is a strong heir over M extending p(xΛ).
β
Let S be a product of some sorts of the language, possibly unboundedly many with repetitions allowed, and X a β
-type-definable subset of S. Let lSβ be the number of factors in S.
Directly from the definition of the content of a tuple of types we get
Remark 3.4*.*
For every natural number n there are only boundedly many possibilities for the content c(p1β,β¦,pnβ) of types p1β,β¦,pnββSSβ(C). More precisely, the number of possible contents is bounded by 2lSβ+2β£Tβ£.
The relation with the Ellis semigroup is given by the following
Proposition 3.5**.**
Let (q1β,β¦,qnβ) and (p1β,β¦,pnβ) be tuples of types from SXβ(C).
Then c(q1β,β¦,qnβ)βc(p1β,β¦,pnβ) if and only if there is Ξ·βEL(SXβ(C)) such that Ξ·(piβ)=qiβ for all i=1,β¦,n.
Proof.
(β) Consider any Ο1β(xΛ,bΛ)βq1β,β¦,Οnβ(xΛ,bΛ)βqnβ. By assumption, there is a tuple bΛβ²β‘β
βbΛ such that Οiβ(xΛ,bΛβ²)βpiβ for all i=1,β¦,n. Take ΟΟ1β(xΛ,bΛ),β¦,Οnβ(xΛ,bΛ)ββAut(C) mapping bΛβ² to bΛ.
Choose a subnet (Οjβ) of the net (ΟΟ1β(xΛ,bΛ),β¦,Οnβ(xΛ,bΛ)β) which converges to some Ξ·βEL. Then Ξ·(piβ)=qiβ for all i.
(β) Consider any (Ο1β(xΛ,yΛβ),β¦,Οnβ(xΛ,yΛβ),q(yΛβ))βc(q1β,β¦,qnβ). Then there is bΛβq(C) such that Οiβ(xΛ,bΛ)βqiβ for all i=1,β¦,n. By the fact that Ξ· is approximated by automorphisms of C, we get ΟβAut(C) such that Οiβ(xΛ,bΛ)βΟ(piβ), and so Οiβ(xΛ,Οβ1(bΛ))βpiβ, holds for all i=1,β¦,n. This shows that (Ο1β(xΛ,yΛβ),β¦,Οnβ(xΛ,yΛβ),q(yΛβ))βc(p1β,β¦,pnβ).
β
This allows us to give a description of all point-transitive subflows of SXβ(C). Given a subflow YβSXβ(C) and qβY with dense orbit, let cYβ=c(q). By the previous proposition, this does not depend on the choice of q. The mapping Yβ¦cYβ is injective and preserves inclusion. In particular, we deduce that there are boundedly many subflows of SXβ(C) (at most βΆ2β(lSβ+2β£Tβ£)). Likewise, there are boundedly many subflows of SXβ(C)n (the n-th Cartesian power of SXβ(C)).
Analogous to the way non-forking can be defined in stable theories as extensions maximal in the fundamental order, we can use the content to define a notion of free extension.
Definition 3.6**.**
Let p(xΛ)βS(A). Say that an extension pβ²(xΛ)βS(C) of p is free if cAβ(pβ²) is minimal among cAβ(r) for r(xΛ)βSpβ(C):={q(xΛ)βS(C):p(xΛ)βq(xΛ)}.
Lemma 3.7**.**
Let p(xΛ)βS(A). Then p has a free extension pβ²(xΛ)βS(C).
Proof.
Expand the language by constants for the elements of A. Let Y be a minimal subflow of Spβ(C) and take pβ²βY (so pβ² is an almost periodic type of the Aut(C)-flow Spβ(C)). Then, by the discussion above, cYβ is minimal among cZβ, ZβSpβ(C) a point-transitive subflow, and hence pβ² is a free extension of p.
β
This provides us with a notion of freeness that is well-defined and satisfies existence in any theory.
One can check (using definability of types) that in the case of a
stable theory it coincides with non-forking (we will not use this in
this paper).
Its properties will be investigated in a future work.
4. Boundedness and absoluteness of the Ellis group
This section is devoted to the proof of Theorem 0.2: in the first part, we will prove boundedness of the Ellis group and we will give an explicit bound on its size; in the second part, we will prove absoluteness. The key tool in both parts is the notion of content of a sequence of types introduced in the previous section. Boundedness will follow easily. The proof of absoluteness is more technical.
Let C be a monster model of an arbitrary theory T, and ΞΊ
the degree of saturation of C. Let S be a product of some number
sorts (posibly unbounded, with repetitions allowed), and let X be a β
-type-definable subset of S.
In this section, by EL we will denote the Ellis semigroup EL(SXβ(C)). Let lSβ be the length of S (i.e. the number of factors in the product S).
We first prove boundedness of the Ellis group. By Remark 3.4, we can find a subset PββnβΟβSXβ(C)n of size β€2lSβ+2β£Tβ£) such that for every n,
[TABLE]
Define R to be the closure of the collection Pprojβ of all types pβSXβ(C) for which there is (p1β,β¦,pnβ)βP such that p=piβ for some i. Note that β£Rβ£β€βΆ2β(β£Pprojββ£)β€βΆ3β(lSβ+2β£Tβ£).
Lemma 4.1**.**
Assume SβSXβ(C) is closed and for every finite tuple pΛβ=(p1β,β¦,pnβ) of types form SXβ(C) there is a finite tuple
qΛβ=(q1β,β¦,qnβ) of types from S with c(qΛβ)βc(pΛβ). Then there exists Ξ·βEL with Im(Ξ·)βS. In particular, there is Ξ·βEL with Im(Ξ·)βR.
Proof.
By Proposition 3.5, for every
finite tuple pΛβ=(p1β,β¦,pnβ) of types from SXβ(C) there
exists Ξ·pΛβββEL such that Ξ·pΛββ(piβ)βS for
all i=1,β¦,n. The collection of all finite tuples of types from
SXβ(C) forms a directed set (where pΛββ€qΛβ if and only
if pΛβ is a subtuple of qΛβ). So the elements Ξ·pΛββ (where pΛβ ranges over all finite tuples of types from
SXβ(C)) form a net which has a subnet convergent to some Ξ·βEL. Then Im(Ξ·)βS, because S is closed. The last
part of the lemma follows by the choice of R.
β
From now on, let M be a minimal left ideal of EL. So
M is partitioned into groups of the form uM,
where uβM ranges over idempotents of M. The
next lemma clarifies the nature of this partition.
Lemma 4.2**.**
Assume u,uβ²βM are idempotents and
I=Im(u),Β Iβ²=Im(uβ²).
uM={hβM:Im(h)=I}.
If uξ =uβ², then Iξ βIβ².
For every hβuM, hβ£Iβ is a permutation of
I. Let Sym(I) denote the group of permutations of I. The
function F:uMβSym(I) mapping h to hβ£Iβ is a
group monomorphism.
For every hβM there is a unique idempotent
uβM wuch that hM=uM. In
particular, hM is the Ellis group of the flow SXβ(C).
Proof.
First notice that for hβuM we have h=uh and u=hhβ²,
where hβ²βuM is the group inverse of H. Hence,
I=Im(u)βIm(h)βI and we get β in (1).
For (2), suppose for a contradiction that IβIβ². Then uβ²u=u
belongs to the group uβ²M, a contradiction.
For β in (1) notice that if hβMβuM, then by (2) and β in (1), Im(h)ξ =I.
(3) Let hβuM. Then there is hβ²βuM with
u=hhβ²=hβ²h. Hence,
[TABLE]
Thus, hIββSym(I). Obviously, F is a group homomorphism. It is
injective, since u is the only idempotent in uM.
(4) is immediate, since hM=huM=uM,
where uβM is the unique idempotent with hβuM.
β
Lemma 4.3**.**
For every Ξ·βEL there is an idempotent uβM such that Im(u)βIm(Ξ·).
Proof.
Take any hβ²βM. Then h:=Ξ·hβ² belongs to M and satisfies Im(h)βIm(Ξ·). By Lemma 4.2(4), there is an idempotent uβhM. Then uβM and Im(u)βIm(Ξ·).
β
We consider the set of functions RR as a semigroup, with
composition of functions. By Lemmas 4.1 and 4.3, we can find an idempotent uβM
with Im(u)βR. Let I=Im(u). By Lemma 4.2(3),
we get the following corollary, which yields the first part of Theorem 0.2, namely to the boundedness of the Ellis group.
Corollary 4.4**.**
The function F:uMβSym(I) given by F(h)=hβ£Iβ is a
group monomorphism. In particular, the size of the Ellis group of the
flow SXβ(C) is bounded by β£Sym(I)β£β€β£Rβ£β£Rβ£, which in turn
is bounded by βΆ4β(lSβ+2β£Tβ£). In the case where lSββ€2β£Tβ£, this bound equals βΆ5β(β£Tβ£).
Proof.
This first part follows directly from Lemma 4.2(3)
The precise bound on the size of the Ellis group follows from the bound on β£Rβ£ computed before Lemma 4.1.
β
By Proposition
2.6 and Corollary 4.4, we get the following corollary which in particular contains the first part of Theorem 0.2 and answers Question 0.1(i).
Corollary 4.5**.**
Let S be a product of an arbitrary (possibly unbounded) number of sorts (with repetitions allowed) and X be a β
-type-definable subset of S. Let cΛ be an enumeration of C and Ξ±Λ a short tuple in C. Then the sizes of the Ellis groups of the flows SXβ(C), ScΛβ(C) and SΞ±Λβ(C) are all bounded by βΆ5β(β£Tβ£).
The following problem is left for the future.
Problem 4.6**.**
Find the optimal (i.e. smallest) upper bound on the size of the Ellis groups of the flows of the form SXβ(C).
In Subsection 7.2, we will see that under the assumption that T has NIP, the set R in Corollary 4.4 can be replaced by the set of types invariant over a fixed small model, which gives us a smaller bound on the size of the Ellis group, namely βΆ3β(β£Tβ£). On the other hand, we know that the optimal bound is at least 2β£Tβ£, as by the material recalled in the final part of Section 1, the size of the Ellis group of ScΛβ(C) is at least β£GalLβ(T)β£ which can be equal to 2β£Tβ£ (e.g. for the theory of countably many independent equivalence relations each of which has two classes).
Now, we turn to the second, more complicated, part of Theorem 0.2, namely the absoluteness of the isomorphism type of the Ellis group of SXβ(C). So from now on, fix any monster models C1ββ»C2β of the theory T. Let EL1β=EL(SXβ(C1β)) and EL2β=EL(SXβ(C2β)), and let M1β and M2β be minimal left ideals of EL1β and EL2β, respectively. Let Ο12β:SXβ(C1β)βSXβ(C2β) be the restriction map.
The general idea is to find
idempotents uβM1β and vβM2β such that Im(u) is of bounded size and Ο12β restricted to Im(u) is a homeomorphism onto Im(v). Having this, we will easily show that the restriction maps F1β:uM1ββSym(Im(u)) and F2β:vM2ββSym(Im(v)) are group monomorphisms, and that Ο12β induces an isomorphism between Im(F1β) and Im(F2β). This implies that uM1ββ
vM2β. In Subsection 7.2, we will see that under NIP such idempotents u and v may be chosen so that their images lie in the set of types invariant over a given model MβΊC. In general, we have to find an abstract substitute of the set of invariant types, which will be a more carefully chosen set R as above. This is done in Corollary 4.14 which requires proving some technical lemmas concerning content. If the reader wishes first to see the main steps of the proof of the main result, he or she can go directly to the statement of Corollary 4.14 and continue reading from that point on.
For types p1β,β¦,pnββSSβ(M), in order to emphasize the model in which we are working, the content c(p1β,β¦,pnβ) of the tuple (p1β,β¦,pnβ) will be denoted by cM(p1β,β¦,pnβ).
Lemma 4.7**.**
Let MβΊN be any models of T and assume that M is β΅0β-saturated. Then each type pβSSβ(M) can be extended to a type pNβSSβ(N) so that for every p1β,β¦,pnββSSβ(M) one has cM(p1β,β¦,pnβ)=cN(p1Nβ,β¦,pnNβ).
Proof.
Let pΛβ=(piβ)iβIβ be an enumeration of the types in SSβ(M) and let aΛβC realize pΛβ. Choose aΛβ² so that tp(aΛβ²/N) extends tp(aΛ/M) and is a strong heir over M. Then the types piNβ:=tp(aΛiβ²β/N) have the required property.
β
From now on, in this section, we assume that S is a product of
boundedly many sorts, unless stated otherwise.
Lemma 4.8**.**
Let M be any small (i.e. of cardinality less than ΞΊ), β΅0β-saturated model of T, and let Y be any subset of SSβ(M). Then there exists a small,
β΅0β-saturated
model Nβ»M and an extension pNβSSβ(N) of each pβY such that:
- (1)
for every p1β,β¦,pnββY one has cM(p1β,β¦,pnβ)=cN(p1Nβ,β¦,pnNβ), and
2. (2)
for every β΅0β-saturated Nβ²β»N and for every
pNβ²βSSβ(Nβ²) extending pN in such a way that cM(p1β,β¦,pnβ)=cNβ²(p1Nβ²β,β¦,pnNβ²β) for all p1β,β¦,pnββY, the restriction map Ο:cl({pNβ²:pβY})βcl({pN:pβY}) is a homeomorphism.
Proof.
Suppose this is not the case. Then
we can construct sequences (NΞ±β)Ξ±<ΞΊβ and (pΞ±β)Ξ±<ΞΊβ for each pβY, where NΞ±ββ»M
is β΅0β-saturated
and pβpΞ±ββSSβ(NΞ±β), such that:
- (i)
β£NΞ±ββ£β€β£Ξ±β£+2β£Tβ£+β£Mβ£,
2. (ii)
Ξ±<Ξ² implies NΞ±ββΊNΞ²β,
3. (iii)
Ξ±<Ξ² implies pΞ±ββpΞ²β,
4. (iv)
for every Ξ± one has cNΞ±β(p1βΞ±β,β¦,pnβΞ±β)=cM(p1β,β¦,pnβ) for all p1β,β¦,pnββY,
5. (v)
for every Ξ± the restriction map ΟΞ±β:cl({pΞ±+1β:pβY})βcl({pΞ±β:pβY}) is not injective.
Then β£cl({pΞ±β:pβY})β£β₯β£Ξ±β£. Taking Ξ±
with β£Ξ±β£>βΆ3β(lSβ+β£Tβ£+β£Mβ£), we get a contradiction to the fact that β£cl({pΞ±β:pβY})β£β€βΆ2β(β£SSβ(M)β£)β€βΆ3β(lSβ+β£Tβ£+β£Mβ£).
β
The next remark follows from Lemma 4.7.
Remark 4.9*.*
If an (β΅0β-saturated) model N satisfies the conclusion of Lemma 4.8, then so does every β΅0β-saturated elementary extension Nβ² of N; this is witnessed by arbitrarily chosen pNβ²βSSβ(Nβ²) for pβY so that pNβpNβ² and cM(p1β,β¦,pnβ)=cNβ²(p1βNβ²,β¦,pnβNβ²) for all p,p1β,β¦,pnββY.
It follows in particular that any β£Nβ£-saturated model extending M satisfies the conclusion of the lemma.
Lemma 4.10**.**
Let N be any small, β΅0β-saturated model of T, and let Z be any subset of SSβ(N). Then there exists a small, β΅0β-saturated model Nβ²β»N and an extension pNβ²βSSβ(Nβ²) of each pβZ such that:
- (1)
for every p1β,β¦,pnββZ one has cN(p1β,β¦,pnβ)=cNβ²(p1Nβ²β,β¦,pnNβ²β), and
2. (2)
for every β΅0β-saturated Nβ²β²β»Nβ² and for every pNβ²β²βSSβ(Nβ²β²) extending pNβ² (where pβZ) in such a way that cN(p1β,β¦,pnβ)=cNβ²β²(p1Nβ²β²β,β¦,pnNβ²β²β) for all p1β,β¦,pnββZ, for every q1β,β¦,qnββcl({pNβ²β²:pβZ}) one has cNβ²β²(q1β,β¦,qnβ)=cNβ²(q1ββ£Nβ²β,β¦,qnββ£Nβ²β).
Proof.
Suppose this lemma is false. Then
we can construct sequences (NΞ±β)Ξ±<ΞΊβ and (pΞ±β)Ξ±<ΞΊβ for each pβZ, where NΞ±ββ»N
is β΅0β-saturated and pβpΞ±ββSSβ(NΞ±β), such that:
- (i)
β£NΞ±ββ£β€β£Ξ±β£+2β£Tβ£+β£Nβ£,
2. (ii)
Ξ±<Ξ² implies NΞ±ββΊNΞ²β,
3. (iii)
Ξ±<Ξ² implies pΞ±ββpΞ²β,
4. (iv)
for every Ξ± one has cNΞ±β(p1βΞ±β,β¦,pnβΞ±β)=cN(p1β,β¦,pnβ) for all p1β,β¦,pnββZ,
5. (v)
for every Ξ± there is n and there are types q1β,β¦,qnββcl({pΞ±+1β:pβZ}) such that cNΞ±β(q1ββ£NΞ±ββ,β¦,qnββ£NΞ±ββ)βcNΞ±+1β(q1β,β¦,qnβ).
For each pβZ, let pβ²=βΞ±βpΞ±ββSSβ(βΞ±βNΞ±β). Since β£cl({pβ²:pβZ})β£β€βΆ2β(β£Zβ£)β€βΆ3β(lSβ+β£Tβ£+β£Nβ£)<ΞΊ and the restriction map from SSβ(βΞ±βNΞ±β) to SSβ(NΞ±β) maps cl({pβ²:pβZ}) onto cl({pΞ±β:pβZ}), we can find a natural number n and types q1β,β¦,qnββcl({pβ²:pβZ}) such that cNΞ±β(q1ββ£NΞ±ββ,β¦,qnββ£NΞ±ββ)βcNΞ±+1β(q1ββ£NΞ±+1ββ,β¦,qnββ£NΞ±+1ββ)
for more than 2lSβ+2β£Tβ£ ordinals Ξ±. But this contradicts Remark 3.4.
β
The next remark follows from Lemma 4.7.
Remark 4.11*.*
If an (β΅0β-saturated) model Nβ² satisfies the conclusion of Lemma 4.10, then so does every β΅0β-saturated elementary extension of Nβ².
By Lemmas 4.8, 4.10 and Remark 4.9, we get
Corollary 4.12**.**
Let M be any small, β΅0β-saturated model of T, and let Y be any subset of SSβ(M). Then there exists a small, β΅0β-saturated
model Nβ²β»M and an extension pNβ²βSSβ(Nβ²) of each pβY such that:
- (1)
for every p1β,β¦,pnββY one has cM(p1β,β¦,pnβ)=cNβ²(p1Nβ²β,β¦,pnNβ²β),
2. (2)
for every β΅0β-saturated Nβ²β²β»Nβ² and for every pNβ²β²βSSβ(Nβ²β²) extending pNβ² in such a way that cM(p1β,β¦,pnβ)=cNβ²β²(p1Nβ²β²β,β¦,pnNβ²β²β) for all p1β,β¦,pnββY, the restriction map Ο:cl({pNβ²β²:pβY)})βcl({pNβ²:pβY}) is a homeomorphism,
3. (3)
for every β΅0β-saturated Nβ²β²β»Nβ² and for every pNβ²β²βSSβ(Nβ²β²) extending pNβ² in such a way that cM(p1β,β¦,pnβ)=cNβ²β²(p1Nβ²β²β,β¦,pnNβ²β²β) for all p1β,β¦,pnββY, for every q1β,β¦,qnββcl({pNβ²β²:pβY}) one has cNβ²β²(q1β,β¦,qnβ)=cNβ²(q1ββ£Nβ²β,β¦,qnββ£Nβ²β).
Proof.
First, we apply Lemma 4.8 to get an β΅0β-saturated model Nβ»M and types pNβSSβ(N) (for all pβY) satisfying the conclusion of Lemma 4.8. Next, we apply Lemma 4.10 to this model N and the set Z:={pN:pβY}, and we obtain an β΅0β-saturated model Nβ²β»N and types pNβ²βSSβ(Nβ²) (for all pβY) satisfying the conclusion of Lemma 4.10; in particular, pNβpNβ² and cM(p1β,β¦,pnβ)=cNβ²(p1Nβ²β,β¦,pnNβ²β) for all p,p1β,β¦,pnββY. By Remark 4.9, Nβ² also satisfies the conclusion of Lemma 4.8 which is witnessed by the types pNβ² for pβY. Therefore, the model Nβ² and the types pNβ² (for pβY) satisfy the conclusion of Corollary 4.12.
β
As above, Lemma 4.7 yields
Remark 4.13*.*
If an (β΅0β-saturated) model Nβ² satisfies the conclusion of Corollary 4.12, then so does every β΅0β-saturated elementary extension of Nβ².
The above technical lemmas and remarks are needed only to prove the following corollary. Recall that X is a β
-type-definable subset of S.
Corollary 4.14**.**
There exist closed, bounded (more precisely, of cardinality less than the degree of saturation of C2β) subsets RC1βββSXβ(C1β) and RC2βββSXβ(C2β) with the following properties.
- (1)
The restriction map Ο12β:SXβ(C1β)βSXβ(C2β) induces a homeomorphism from RC1ββ onto RC2ββ, which will also be denoted by Ο12β.
2. (2)
For every n, {cC2β(p1β,β¦,pnβ):p1β,β¦,pnββSXβ(C2β)}={cC2β(p1β,β¦,pnβ):p1β,β¦,pnββRC2ββ}.
3. (3)
For every n, cC1β(p1β,β¦,pnβ)=cC2β(Ο12β(p1β),β¦,Ο12β(pnβ)) for all p1β,β¦,pnββRC1ββ.
4. (4)
For every n and for every p1β,β¦,pnββSXβ(C1β) there exist q1β,β¦,qnββRC1ββ such that cC1β(q1β,β¦,qnβ)βcC1β(p1β,β¦,pnβ).
Proof.
As at the beginning of Section 4, by Remark 3.4,
we can find a subset PββnβΟβSXβ(C2β)n of bounded size (in fact, of size β€2lSβ+2β£Tβ£) such that for every n
[TABLE]
Let Pprojβ be the collection of all types pβSXβ(C2β) for which there is (p1β,β¦,pnβ)βP such that p=piβ for some i.
Since Pprojβ is of bounded size, there is a small, β΅0β-saturated MβΊC2β such that for every n and for every p1β,β¦,pnββPprojβ, cC2β(p1β,β¦,pnβ)=cM(p1ββ£Mβ,β¦,pnββ£Mβ). Put
[TABLE]
Now, take a small, β΅0β-saturated model Nβ²β»M provided by Corollary 4.12. We can assume that Nβ²βΊC2β.
By Remark 4.13, the model C2β in place of Nβ² also satisfies the properties in Corollary 4.12 (here C2β is small with respect to C1β). So we get types pC2ββSXβ(C2β) (for all pβY) satisfying properties (1)-(3) from Corollary 4.12 and such that pβpC2β for all pβY. In particular:
- (i)
for every p1β,β¦,pnββY one has cM(p1β,β¦,pnβ)=cC2β(p1C2ββ,β¦,pnC2ββ),
2. (ii)
for every pC1ββSSβ(C1β) extending pC2β in such a way that cM(p1β,β¦,pnβ)=cC1β(p1C1ββ,β¦,pnC1ββ) for all p1β,β¦,pnββY, the restriction map Ο:cl({pC1β:pβY)})βcl({pC2β:pβY}) is a homeomorphism,
3. (iii)
for every pC1ββSSβ(C1β) extending pC2β in such a way that cM(p1β,β¦,pnβ)=cC1β(p1C1ββ,β¦,pnC1ββ) for all p1β,β¦,pnββY, for every q1β,β¦,qnββcl({pC1β:pβY}) one has cC1β(q1β,β¦,qnβ)=cC2β(q1ββ£C2ββ,β¦,qnββ£C2ββ).
By (i) and Lemma 4.7 applied to the models C2ββΊC1β, for each pβY we can find pC1ββSXβ(C1β) extending pC2β and such that cM(p1β,β¦,pnβ)=cC1β(p1C1ββ,β¦,pnC1ββ) for all p1β,β¦,pnββY.
Define
[TABLE]
We will check now that these sets have the desired properties.
Properties (1) and (3) follow from (ii) and (iii), respectively. Property (2) follows from (i) and the choice of P and M. Property (4) follows from (1)-(3), but we will check it. Consider any p1β,β¦,pnββSXβ(C1β). Clearly, cC1β(p1β,β¦,pnβ)βcC2β(p1ββ£C2ββ,β¦,pnββ£C2ββ). By (2), we can find r1β,β¦,rnββRC2ββ such that cC2β(p1ββ£C2ββ,β¦,pnββ£C2ββ)=cC2β(r1β,β¦,rnβ). By (1), define q1β:=Ο12β1β(r1β),β¦,qnβ:=Ο12β1β(rnβ). Then, by (3), cC2β(r1β,β¦,rnβ)=cC1β(q1β,β¦,qnβ). We conclude that cC1β(q1β,β¦,qnβ)βcC1β(p1β,β¦,pnβ).
β
In the rest of this section we take the notation from Corollary 4.14. Property (1) will be used many times without mentioning. From now on, Ο12β denotes the homeomorphism from RC1ββ to RC2ββ.
Lemma 4.15**.**
For every fβEL1β such that f[RC1ββ]βRC1ββ there exists gβEL2β such that g[RC2ββ]βRC2ββ and Ο12ββfβ£RC1βββ=gβΟ12ββ£RC1βββ.
Proof.
Consider any pairwise distinct types p1β,β¦,pnββRC2ββ and any formulas
[TABLE]
Let pΛβ=(p1β,β¦,pnβ), ΟΛβ=(Ο1β(xΛ,yΛβ),β¦,Οnβ(xΛ,yΛβ)), and ΟΛβaΛβ=(Ο1β(xΛ,aΛ),β¦,Οnβ(xΛ,aΛ)).
We claim that it is enough to find ΟpΛβ,ΟΛβaΛβββAut(C2β) such that
[TABLE]
Indeed, taking the partial directed order on all such pairs (pΛβ,ΟΛβaΛβ) (where formulas are treated as elements of the Lindenbaum algebra) given by
[TABLE]
the net (ΟpΛβ,ΟΛβaΛββ) has a subnet convergent to some gβEL2β which satisfies the conclusion.
Now, we will explain how to find ΟpΛβ,ΟΛβaΛββ. By the choice of the Οiββs, there exists ΟpΛβ,ΟΛβaΛβββAut(C1β) such that
[TABLE]
This implies that for q:=tp(aΛ/β
) one has (Ο1β,β¦,Οnβ,q)βcC1β(Ο12β1β(pΛβ)).
Hence, by Property (3) in Corollary 4.14, (Ο1β,β¦,Οnβ,q)βcC2β(pΛβ). This means that there exists ΟpΛβ,ΟΛβaΛββ²ββAut(C2β) such that
[TABLE]
So, ΟpΛβ,ΟΛβaΛββ:=ΟpΛβ,ΟΛβaΛββ²β1β is as required.
β
Lemma 4.16**.**
For every gβEL2β such that g[RC2ββ]βRC2ββ there exists fβEL1β such that f[RC1ββ]βRC1ββ and Ο12ββfβ£RC1βββ=gβΟ12ββ£RC1βββ.
Proof.
We proceed as in Lemma 4.15. Consider
any pairwise distinct types p1β,β¦,pnββRC1ββ and any
formulas
[TABLE]
As in Lemma 4.15, it is enough to find
Ο=ΟpΛβ,ΟΛβaΛβββAut(C1β) such
that Οiβ(xΛ,aΛ)βΟ(piβ) for i=1,β¦,n, where
pΛβ,Β ΟΛβaΛβ are as in the proof there.
By property (3) from Corollary 4.14, cC2β(gβΟ12β(pΛβ))=cC1β(Ο12β1ββgβΟ12β(pΛβ)), hence we
find aΛβ²βC2β with aΛβ²β‘β
βaΛ
and Οiβ(xΛ,aΛβ²)βgβΟ12β(piβ) for
i=1,β¦,n. Then there is ΟβAut(C2β) with
Οiβ(xΛ,aΛβ²)βΟ(Ο12β(piβ)) for i=1,β¦,n.
Let aΛβ²β²=Οβ1(aΛβ²). Hence, aΛβ²β²β‘β
βaΛβ²β‘β
βaΛ and
Οiβ(xΛ,aΛβ²β²)βΟ12β(piβ)βpiβ. Therefore,
there is ΟβAut(C1β) with Ο(aΛβ²β²)=aΛ. Clearly, Ο satisfies our demands.
β
Recall that M1β and M2β are minimal left ideals of EL1β and EL2β, respectively.
By Properties (4) and (2) in Corollary 4.14 and Lemmas 4.1 and 4.3, we get idempotents uβM1β and vβ²βM2β such that Im(u)βRC1ββ and Im(vβ²)βRC2ββ. Note that for any such idempotents Im(u)=Im(uβ£RC1βββ) and Im(vβ²)=Im(vβ²β£RC2βββ).
Lemma 4.17**.**
There exists an idempotent vβM2β such that Ο12β[Im(u)]=Im(v).
Proof.
By Lemma 4.15, there is gβEL2β such that Ο12ββuβ£RC1βββ=gβΟ12ββ£RC1βββ. Then Ο12β[Im(u)]=Im(gβ£RC2βββ). Let h=gvβ²βM2β. By Lemma 4.2(4), there is an idempotent vβhM2β such that hM2β=vM2β. We will show that Ο12β[Im(u)]=Im(v).
We clearly have
[TABLE]
hence also
[TABLE]
By Lemma 4.16, there is fβEL1β such
that fβ£RC1βββ=Ο12β1ββvβΟ12ββ£RC1βββ. Then
Im(fβ£RC1βββ)=Ο12β1β[Im(v)]. Choose an idempotent
uβ²βM1β with fuβuβ²M1β. Then, using
(3) and Lemma 4.2, we have
[TABLE]
By Lemma 4.2, u=uβ² and
Im(u)=Im(uβ²)=Ο12β1β[Im(v)]. Hence, Im(v)=Ο12β[Im(u)].
β
Take v from the above lemma. Let I1β:=Im(u)βRC1ββ and
I2β:=Im(v)βRC2ββ. Then, for every fβuM1β and gβvM2β we have Im(f)=I1β and
Im(g)=I2β. By Lemma 4.2(3), we get group
monomorphisms F1β:uM1ββSym(I1β) and F2β:vM2ββSym(I2β). By the choice of v in Lemma 4.17, Ο12β induces an isomorphism of
topological groups Ο12β²β:Sym(I1β)βSym(I2β).
Lemma 4.18**.**
Ο12β²β[Im(F1β)]=Im(F2β).
Proof.
(β) Take fβuM1β. By Lemma 4.15, we can find gβEL2β such that Ο12ββfβ£RC1βββ=gβΟ12ββ£RC1βββ. Hence, Ο12ββfβ£I1ββ=gβΟ12ββ£I1ββ. By the choice of v in Lemma 4.17 and the fact that vβ£I2ββ=idI2ββ, this implies that Ο12ββfβ£I1ββ=vgvβΟ12ββ£I1ββ. Note that vgvβvM2β, and we get
[TABLE]
(β) Take gβvM2β. By Lemma 4.16, we can find fβEL1β such that f[RC1ββ]βRC1ββ and Ο12ββfβ£RC1βββ=gβΟ12ββ£RC1βββ. Hence, Ο12ββfβ£I1ββ=gβΟ12ββ£I1ββ. By the choice of v in Lemma 4.17, the fact that uβ£I1ββ=idI1ββ, and the fact that Ο12β:RC1βββRC2ββ is a bijection, this implies that Ο12ββufuβ£I1ββ=gβΟ12ββ£I1ββ. Note that ufuβuM1β, and we get
[TABLE]
As a conclusion, we get
absoluteness of the isomorphism type of the Ellis group.
Corollary 4.19**.**
The map F2β1ββΟ12β²ββF1β:uM1ββvM2β is a group isomorphism. Thus the Ellis group of the flow SXβ(C) does not depend (up to isomorphism) on the choice of the monster model C.
By Proposition 2.6 this implies the second part of Theorem 0.2.
Corollary 4.20**.**
Here, let S be a product of an arbitrary (possibly unbounded) number of sorts and X be a β
-type-definable subset of S. Then the Ellis group of the flow SXβ(C) does not depend (up to isomorphism) on the choice of the monster model C.
By virtue of Corollaries 4.4 and 4.19 together with Proposition 2.6, in order to complete the proof of Theorem 0.2, it remains to show
Proposition 4.21**.**
F2β1ββΟ12β²ββF1β:uM1ββvM2β* is a homeomorphism, where uM1β and vM2β are equipped with the Ο-topology.*
The proof will consist of two lemmas.
Let F1β²β:M1ββSXβ(C1β)I1β be the
restriction map (so it is an extension of F1β to a bigger
domain). As in the case of F1β, we easily get that F1β²β is
injective: if F1β²β(f)=F1β²β(g) for some f,gβM1β
(which implies that f=fu and g=gu), then fβ£I1ββ=gβ£I1ββ,
so, since Im(u)=I1β, we get f=fu=gu=g.
Aut(C1β) acts on EL1β and on M1β; this induces an
action of Aut(C1β) also on Im(F1β²β)βSXβ(C1β)I1β: for ΟβAut(C1β) and
dβM1β define Ο(F1β²β(d)) as F1β²β(Οd)=(Οd)β£I1ββ.
Lemma 4.22**.**
Let DβuM1β. Then F1β[clΟβ(D)] is the set of all fβIm(F1β) for which there exist nets (Οiβ)iβ in Aut(C1β) and (diβ)iβ in D such that limΟiβ²β=idI1ββ (where Οiβ²β is the element of SXβ(C1β)I1β induced by Οiβ) and limΟiβ(F1β(diβ))=f. The same statement also holds for C2β, vM2β, I2β and F2β in place of C1β, uM1β, I1β, and F1β, respectively.
Proof.
(β) Take any gβclΟβ(D) and f=F1β(g). By the definition of the Ο-closure (see [10, Subsection 1.1]), there exist nets (Οiβ)iβ in Aut(C1β) and (diβ)iβ in D such that limΟiβ=u and limΟiβ(diβ)=g. Since uβ£I1ββ=idI1ββ and F1β²β is continuous with respect to the
product topologies on the domain and on the target, we see that limΟiβ²β=idI1ββ and limΟiβ(F1β(diβ))=limΟiβ(F1β²β(diβ))=limF1β²β(Οiβ(diβ))=F1β²β(g)=f.
(β) Consider any fβIm(F1β) for which there exist nets (Οiβ)iβ in Aut(C1β) and (diβ)iβ in D such that limΟiβ²β=idI1ββ and limΟiβ(F1β(diβ))=f.
There is a subnet (Οjβ)jβ of (Οiβ)iβ such that h:=limΟjβ exists in EL1β and for the corresponding subnet (ejβ)jβ of (diβ)iβ the limit g:=limjβΟjβ(ejβ) also exists (and belongs to M1β).
Now, we will use the circle operation β from the definition of the Ο-closure (see [10, Definition 1.3]). To distinguish it from composition of functions (for which the symbol β is reserved in this paper), the circle operation will be denoted by β.
We have that gβhβDβhβ(uβD)β(hu)βD. But hβ£I1ββ=limΟjβ²β=limΟiβ²β=idI1ββ and Im(u)=I1β, so hu=u. Hence, gβuβD. We also see that f=limiβΟiβ(F1β(diβ))=limjβΟjβ(F1β(ejβ))=limjβF1β²β(Οjβ(ejβ))=F1β²β(g). But fβIm(F1β)=F1β²β[uM1β] and F1β²β is injective. Therefore, gβ(uβD)β©uM1β=:clΟβ(D). Hence, f=F1β(g)βF1β[clΟβ(D)].
β
Equip Im(F1β) and Im(F2β) with the topologies induced by the isomorphisms F1β and F2β from the Ο-topologies on uM1β and vM2β, respectively. The corresponding closure operators will be denoted by clΟ1β and clΟ2β. They are given by the right hand side of Lemma 4.22. The next lemma will complete the proof of Proposition 4.21.
Lemma 4.23**.**
For any DβIm(F1β), Ο12β²β[clΟ1β(D)]=clΟ2β(Ο12β²β[D]).
The proof of this lemma will be an elaboration on the proofs of Lemmas 4.15 and 4.16.
Proof.
(β) Take any fβclΟ1β(D). By Lemma 4.22, there exist nets (Οiβ)iβ in Aut(C1β) and (diβ)iβ in D such that limΟiβ²β=idI1ββ (where Οiβ²β is the element of SXβ(C1β)I1β induced by Οiβ) and limΟiβ(diβ)=f.
Consider any pairwise distinct p1β,β¦,pnββI2β and any formulas
[TABLE]
Let pΛβ=(p1β,β¦,pnβ), ΟΛβ=(Ο1β(xΛ,yΛβ),β¦,Οnβ(xΛ,yΛβ)), ΟΛβ=(Ο1β(xΛ,yΛβ),β¦,Οnβ(xΛ,yΛβ)), and ΟΛβaΛβ=(Ο1β(xΛ,aΛ),β¦,Οnβ(xΛ,aΛ)), ΟΛβaΛβ=(Ο1β(xΛ,aΛ),β¦,Οnβ(xΛ,aΛ)).
We claim that it is enough to find ΟpΛβ,ΟΛβaΛβ,ΟΛβaΛβββAut(C2β) and an index ipΛβ,ΟΛβaΛβ,ΟΛβaΛββ such that
[TABLE]
Indeed, taking the partial directed order on all triples (pΛβ,ΟΛβaΛβ,ΟΛβaΛβ) defined as in the proof of Lemma 4.15,
the net (ΟpΛβ,ΟΛβaΛβ,ΟΛβaΛββ) satisfies limΟpΛβ,ΟΛβaΛβ,ΟΛβaΛββ²β=idI2ββ and limΟpΛβ,ΟΛβaΛβ,ΟΛβaΛββ(Ο12β²β(dipΛβ,ΟΛβaΛβ,ΟΛβaΛβββ))=Ο12β²β(f), which by Lemmas 4.22 and 4.18 implies that Ο12β²β(f)βclΟ2β(Ο12β²β[D]).
Now, we will explain how to find ΟpΛβ,ΟΛβaΛβ,ΟΛβaΛββ. By the fact that limΟiβ²β=idI1ββ and limΟiβ(diβ)=f and by the choice of ΟΛβaΛβ and ΟΛβaΛβ, there is an index ipΛβ,ΟΛβaΛβ,ΟΛβaΛββ for which
[TABLE]
Since Ο12β1β(p1β),β¦,Ο12β1β(pnβ)βRC1ββ and dipΛβ,ΟΛβaΛβ,ΟΛβaΛβββ(Ο12β1β(p1β)),β¦,dipΛβ,ΟΛβaΛβ,ΟΛβaΛβββ(Ο12β1β(pnβ))βRC1ββ, the existence of ΟpΛβ,ΟΛβaΛβ,ΟΛβaΛβββAut(C2β) with the required properties follows easily from Property (3) in Corollary 4.14 (similarly to the final part of the proof of Lemma 4.15).
(β) Assume gβclΟ2β(Ο12β²β[D]). Our goal is to
prove that the function f:=Ο12β²β1β(g)=Ο12β1ββgβΟ12ββ£I1ββ belongs to clΟ1β(D).
By Lemma 4.22 there exist nets
(Οiβ)iβ in Aut(C2β) and (diβ)iβ in D such that limΟiβ²β=idI2ββ (where Οiβ²ββSXβ(C2β)I2β is induced
by Οiβ) and limΟiβ(Ο12β²β(diβ))=g. Consider any pairwise
distinct p1β,β¦,pnββI1β and any formulas
[TABLE]
[TABLE]
As in the proof of (β), it is enough to find
Ο=ΟpΛβ,ΟΛβaΛβ,ΟΛβaΛβββAut(C1β) and iβ=ipΛβ,ΟΛβaΛβ,ΟΛβaΛββ such that Οjβ(xΛ,aΛ)βΟ(diββ(pjβ)) and Οjβ(xΛ,aΛ)βΟ(pjβ)
for j=1,β¦,n. We explain how to find Ο and iβ.
We have that pjβ and f(pjβ) belong to I1ββRC1ββ and
Ο12β(f(pjβ))=g(Ο12β(pjβ)). Hence, by Property (3) in
Corollary 4.14, we get that
[TABLE]
Hence, there is aΛβ² in C2β with aΛβ²β‘β
βaΛ such that Οjβ(xΛ,aΛβ²)βg(Ο12β(pjβ)) and Οjβ(xΛ,aΛβ²)βΟ12β(pjβ) for j=1,β¦,n. By the fact that limΟiβ²β=idI2ββ and limΟiβ(Ο12β²β(diβ))=g, there is
an index iβ such that Οjβ(xΛ,aΛβ²)βΟiββ(Ο12β²β(diββ))(Ο12β(pjβ)) and Οjβ(xΛ,aΛβ²)βΟiββ(Ο12β(pjβ)) for j=1,β¦,n.
Let aΛβ²β²=Οiββ1β(aΛβ²). Hence, Οjβ(xΛ,aΛβ²β²)βΟ12β(diββ(pjβ))βdiββ(pjβ) and
Οjβ(xΛ,aΛβ²β²)βΟ12β(pjβ)βpjβ for j=1,β¦,n, and also
aΛβ²β²β‘β
βaΛβ²β‘β
βaΛ. Therefore, there is ΟβAut(C1β) with Ο(aΛβ²β²)=aΛ. Clearly, Ο satisfies our demands.
β
The proof of Proposition 4.21 has been completed. As was mentioned before, Corollaries 4.4, 4.19, Proposition 4.21, and Proposition 2.6 give us Theorem 0.2, which implies immediately the following
Corollary 4.24**.**
Here, let S be a product of an arbitrary (possibly unbounded) number of sorts and X be a β
-type-definable subset of S. Then, the quotient of the Ellis group uM of the flow SXβ(C) by the intersection H(uM) of the Ο-closures of the Ο-neighborhoods of the identity does not depend (as a compact topological group) on the choice of C.
Theorem 0.2 together with Proposition 2.4 imply that the answer to Question 0.1 is positive.
Corollary 4.25**.**
The Ellis groups of the flows ScΛβ(C) and SΞ±Λβ(C) (where cΛ is an enumeration of C and Ξ±Λ is a short tuple in C) are of bounded size and do not depend (as groups equipped with the Ο-topology) on the choice of the monster model C. Thus, the quotients of these groups by the intersections of the Ο-closures of the Ο-neighborhoods of the identity are also absolute as topological groups.
In order for the proof of Theorem 0.2 to
proceed smoothly, we assumed that our monster models are
ΞΊ-saturated and strongly ΞΊ-homogeneous, where ΞΊ
was a strong limit cardinal larger than β£Tβ£. A closer inspection of
the proof shows we may relax the restriction on ΞΊ to assume
just that ΞΊ>βΆ5β(β£Tβ£).
5. Boundedness and absoluteness of minimal left ideals
We will start from an example where the minimal left ideals in the
Ellis semigroup of the flow SXβ(C) (even of SΞ±β(C) where
Ξ±βC) are of unbounded size, answering negatively Question
0.3(i). Then we give a characterization in
terms of Lascar invariant types of when these ideals are of bounded
size, and, if it is the case, we find an explicit bound on the
size. The main part of this section is devoted to the proof of Theorem
0.4. For this we will use strong heir extensions and the aforementioned characterization in terms of Lascar invariant types.
Example 5.1**.**
Consider M=(S1,R(x,y,z)), where S1 is the unit circle on the plane, and R(x,y,z) is the circular order, i.e. R(a,b,c) holds if and only if a,b,c are pairwise distinct and b comes before c going around the circle clockwise starting at a. Let Cβ»M be a monster model. It is well-known that this structure has quantifier elimination, so there is a unique complete 1-type over β
. Moreover, Th(M) has NIP.
S1β(C) is a point-transitive Aut(C)-flow, consisting of
extensions of the unique type in S1β(β
).
Let NA be the set of all non-algebraic types in S1β(C) (they
correspond to cuts in the non-standard S1(C)). Let
C be the set of all constant functions S1β(C)βS1β(C), with
values in NA. We claim that:
- (1)
CβEL(S1β(C)).
2. (2)
EL(S1β(C))Ξ·=C for every Ξ·βC.
3. (3)
C is the unique minimal left ideal in EL(S1β(C)), and it is
unbounded.
4. (4)
The Ellis group of S1β(C) is trivial (so bounded).
5. (5)
The minimal left ideals in EL(ScΛβ(C)) are of unbounded
size, where cΛ is an enumeration of C.
Proof.
(1) For example let qβNA be the right cut at
1. So q is generated by formulas Ο(x,a)=R(1,x,a), where
aβS1(C)β{1}. Consider any p1β,β¦,pnββS1β(C), let pΛβ=(p1β,β¦,pnβ) and let aβS1(C)β{1}. By quantifier elimination and the strong
homogeneity of C, one can easily find ΟpΛβ,Ο(x,a)ββAut(C) such that Ο(x,a)βΟpΛβ,Ο(x,a)β(piβ) for all i=1,β¦,n. Then
(ΟpΛβ,Ο(x,a)β) is a net (with the obvious ordering
on the indexes) which converges to the function Ξ·βEL(S1β(C)) constantly equal to q.
(2) (β) is clear, and (β) follows from (1).
(3) C is a minimal left ideal by (2). It is unbounded as NA is
unbounded. For every Ξ·βC and Ξ·β²βEL(S1β(C)) we have
Ξ·βΞ·β²=Ξ·, hence CβEL(S1β(C))Ξ·β², and so C is
a unique minimal left ideal.
(4) Every element of C is an idempotent.
(5) follows from (3) and Corollary 2.2.
β
Knowing that minimal left ideals may be of unbounded size, a natural goal is to understand when they are of bounded size, and whether boundedness is absolute. So these are our next goals (as to the first goal, a more satisfactory answer is obtained in Section 7 under the NIP assumption).
As usual, let C be a monster model of an arbitrary theory T. Let
S be a product of some number of sorts (possibly unbounded, with repetitions allowed), and let X be a β
-type-definable subset of S. In this section, by EL we will denote the Ellis semigroup EL(SXβ(C)). Let M be a minimal left ideal in EL. By lSβ denote the length of S (i.e. the number of factors in the product S). Let ILβ denote the set of all Lascar invariant types in SXβ(C), i.e. types which are invariant under AutfLβ(C). Note that ILβ is either empty or an Aut(C)-subflow of SXβ(C) (in particular, it is closed) and it is of bounded size.
Proposition 5.2**.**
If G is a closed, bounded index subgroup of Aut(C) (with Aut(C) equipped with the pointwise convergence topology), then AutfLβ(C)β€G.
Proof.
Let Οiβ, i<Ξ», by a set of representatives of right costs of G in Aut(C) (so Ξ» is bounded). Then
[TABLE]
is a closed, normal, bounded index subgroup of Aut(C).
Consider any ΟβAutfLβ(C) and any finite aΛβC. The orbit equivalence relation of Gβ² on the set of
realizations of tp(aΛ/β
) is a bounded invariant
equivalence relation, so it is coarser than ELβ. Therefore, there
exists ΟaΛββGβ² with ΟaΛβ(aΛ)=Ο(aΛ).
This shows that Ο lies in the closure of Gβ², and hence ΟβGβ² (as Gβ² is closed). In this way, we proved that AutfLβ(C)β€Gβ²β€G.
β
Remark 5.3*.*
For every pβSXβ(C), StabAut(C)β(p):={ΟβAut(C):Ο(p)=p} is a closed subgroup of Aut(C).
The next corollary follows immediately from Proposition 5.2 and Remark 5.3.
Corollary 5.4**.**
Let pβSXβ(C). Then the orbit Aut(C)p is of bounded size if and only if pβILβ.
The next proposition is our characterization of boundedness of the minimal left ideals, and it yields an explicit bound on their size.
Proposition 5.5**.**
*The following conditions are equivalent.
i) The minimal left ideal M is of bounded size.
ii) For every Ξ·βM the image Im(Ξ·) is contained in ILβ.
iii) For some Ξ·βEL the image Im(Ξ·) is contained in ILβ.
Moreover, if M is of bounded size, then β£Mβ£β€βΆ3β(β£Tβ£).*
Proof.
Suppose first that M is of bounded size. Then for every Ξ·βM and for every pβSXβ(C) the orbit Aut(C)Ξ·(p) is of bounded size, so Ξ·(p)βILβ in virtue of Corollary 5.4.
Now, suppose that for some Ξ·βEL the image Im(Ξ·) is contained in ILβ. This means that the size of the orbit Aut(C)Ξ· is bounded by β£GalLβ(T)β£β€2β£Tβ£, and so β£cl(Aut(C)Ξ·)β£β€βΆ3β(β£Tβ£). So the size of M is bounded by βΆ3β(β£Tβ£), because it is equal to the size of a minimal Aut(C)-subflow
contained in cl(Aut(C)Ξ·).
β
The next corollary will be used later.
Corollary 5.6**.**
The minimal left ideal M is of bounded size if and only if for every types q1β,β¦,qnββSXβ(C) and formulas Ο1β(xΛ,yΛβ1β),β¦,Οnβ(xΛ,yΛβnβ), for every (aΛ1β,bΛ1β)βELβ,β¦,(aΛnβ,bΛnβ)βELβ there exists ΟβAut(C) such that
[TABLE]
Proof.
Suppose first that M is of bounded size. Take any Ξ·βM. Then, by Proposition 5.5, for every qβSXβ(C), Ξ·(q) is Lascar invariant. So, for any data as on the right hand side of the required equivalence we have that Οiβ(xΛ,aΛiβ)β§Β¬Οiβ(xΛ,bΛiβ)β/Ξ·(qiβ) for i=1,β¦,n. Hence, since Ξ· is approximated by elements of Aut(C), there exists ΟβAut(C) with the required property.
Now, assume that the right hand side holds.
The Ο from our assumptions depends on qΛβ:=(q1β,β¦,qnβ),Β ΟΛβ:=(Ο1β,β¦,Οnβ),Β aΛ:=(aΛ1β,β¦,aΛnβ) and bΛ:=(bΛ1β,β¦,bΛnβ). So we may write
Ο as ΟqΛβ,ΟΛβ,aΛ,bΛβ. We order the
set of indexes (qΛβ,ΟΛβ,aΛ,bΛ) by:
[TABLE]
if and only if (qiβ,Οiβ,aΛiβ,bΛiβ)iβ€mβ is a subsequence of (qiβ²β,Οiβ²β,aΛiβ²β,bΛiβ²β)iβ€nβ.
Taking the limit of a convergent subnet of the net (ΟqΛβ,ΟΛβ,aΛ,bΛβ1β),
we get an element Ξ·β²βEL such that Im(Ξ·β²)βILβ.
Therefore, M is of bounded size by Proposition 5.5.
β
From now on, fix any monster models C1ββ»C2β of the theory T. (For the purpose of our main results, without loss of generality, we can always assume that C1β is a monster model with respect to the size of C2β.) Let EL1β=EL(SXβ(C1β)) and EL2β=EL(SXβ(C2β)).
By IL,Ciββ denote the set of all Lascar invariant types in SXβ(Ciβ), for i=1,2.
The following remark is folklore.
Remark 5.7*.*
Let Ο12β:SXβ(C1β)βSXβ(C2β) be the restriction
map. Then Ο12ββ£IL,C1βββ:IL,C1βββIL,C2ββ
is a homeomorphism.
(For pβIL,C2ββ, (Ο12ββ£IL,C1βββ)β1(p) is the
unique M-invariant extension of p to a type in SXβ(C1β), where
MβΊC2β is small.)
From now on, Ο12β will denote the above homeomoprhism from IL,C1ββ to IL,C2ββ.
Proposition 5.8**.**
The minimal left ideals in EL1β are of bounded size if and only if
the minimal left ideals in EL2β are of bounded size.
Proof.
(β) Suppose the minimal left ideals in EL2β are of
unbounded size. By Corollary 5.6, there are types q1β,β¦,qnββSXβ(C2β), formulas Ο1β(xΛ,yΛβ1β),β¦,Οnβ(xΛ,yΛβnβ) and tuples (aΛ1β,bΛ1β),β¦,(aΛnβ,bΛnβ)βELβ (in C2β) such that for every ΟβAut(C2β),
[TABLE]
This means that
[TABLE]
where Οiβ(xΛ,yΛβ)=Β¬Οiβ(xΛ,yΛβiβ)β¨Οiβ(xΛ,yΛβiβ²β),Β yΛβ=(yΛβ1β,yΛβ1β²β,β¦,yΛβnβ,yΛβnβ²β) and p(yΛβ)=tp(aΛ1β,bΛ1β,β¦,aΛnβ,bΛnβ). By Lemma 4.7, there are types
q1β²β,β¦,qnβ²ββSXβ(C1β) with
c(q1β,β¦,qnβ)=c(q1β²β,β¦,qnβ²β). The types qiβ²β, formulas
Οiβ and tuples (aΛiβ,bΛiβ),i=1,β¦,n, witness that
the right hand side of Corollary 5.6 fails for C1β. Hence the minimal
left ideals in EL1β are of unbounded size.
(β) Suppose that the minimal left ideals in EL2β are of bounded size.
To deduce that the same is true in EL1β, we have to check that the right hand side of Corollary 5.6 holds for C1β. So consider any q1β,β¦,qnββSXβ(C1β), formulas Ο1β(xΛ,yΛβ1β),β¦,Οnβ(xΛ,yΛβnβ), and tuples (aΛ1β,bΛ1β)βELβ,β¦,(aΛnβ,bΛnβ)βELβ (where aΛiβ corresponds to yΛβiβ), and the goal is to find an appropriate ΟβAut(C1β). Choose a model C2β²ββ
C2β such that C2β²ββΊC1β and aΛiβ,bΛiβ are contained in C2β²β for all i=1,β¦,n. Let q1β²β=q1ββ£C2β²ββ,β¦qnβ²β=qnββ£C2β²ββ. By assumption, the minimal left ideals in EL(SXβ(C2β²β)) are of bounded size, so Corollary 5.6 yields Οβ²βAut(C2β²β) such that
[TABLE]
Then any extension ΟβAut(C1β) of Οβ² does the job.
β
Proposition 5.8 gives us Theorem 0.4(i).
Corollary 5.9**.**
Let S be a product of an arbitrary (possibly unbounded) number of sorts, and X be β
-type-definable subset of S. Then the property that a minimal left ideal of the Ellis semigroup of the flow SXβ(C) is of bounded size is absolute (i.e. does not depend on the choice of C).
By Proposition 5.5, Corollary 5.9, Proposition 2.4, we get
Corollary 5.10**.**
Let Ξ±Λ be a short tuple, and cΛ be an enumeration of C. The property that the minimal left ideals in EL(ScΛβ(C)) [resp. in EL(SΞ±Λβ(C))] are of bounded size is absolute. Moreover, in each of these two cases, if the minimal left ideals are of bounded size, then this size is bounded by βΆ3β(β£Tβ£).
To show the second part of Theorem 0.4, we need some preparatory observations, which explain better the whole picture.
Assume for the rest of this section that ILβ is not empty. Then
ILβ is an Aut(C)-flow, hence it has its own Ellis semigroup EL(ILβ) which is a subflow of the Aut(C)-flow ILILββ.
Proposition 5.11**.**
*Let FΛ:ELβSXβ(C)ILβ be the restriction map. Then:
i) FΛ is a homomorphism of Aut(C)-flows.
ii) Im(FΛ)=EL(ILβ)βILILββ, so from now on we treat FΛ as a function going to EL(ILβ). Then FΛ is an epimorphism of Aut(C)-flows and of semigroups.
iii) FΛ[M] is a minimal subflow (equivalently, minimal left ideal) of EL(ILβ). Let F=FΛβ£Mβ:MβFΛ[M].
iv) If M is of bounded size, then F:MβIm(F) is an isomorphism of Aut(C)-flows and of semigroups.*
Proof.
Point (i) is obvious. Point (ii) follows from (i) and compactness of the spaces in question, namely: Im(FΛ)=FΛ[cl(Aut(C)idSXβ(C)β)]=cl(Aut(C)idILββ)=EL(ILβ). Point (iii) follows from (ii).
Let us show (iv). Take an idempotent uβM. By Proposition 5.5, Im(u)βILβ. We need to show that F is injective. This follows by the same simple argument as in the paragraph following Proposition 4.21: if F(f)=F(g) for some f,gβM (which implies that f=fu and g=gu), then fβ£ILββ=gβ£ILββ, so, since Im(u)βILβ, we get f=fu=gu=g.
β
Coming back to the situation with two monster models C1ββ»C2β, note that each of the spaces IL,C1ββ, IL,C1βSXβ(C1β)β, IL,C1βIL,C1βββ, and EL(IL,C1ββ) is naturally an Aut(C2β)-flow with the action of Aut(C2β) defined by: Οx:=Οβ²x, where Οβ²βAut(C1β) is an arbitrary extension of ΟβAut(C2β). Using the fact that for every ΟβAut(C1β) there is ΟβAut(C1β) which preserves C2β setwise (i.e. which is an extension of an automorphism of C2β) and such that Ο/AutfLβ(C1β)=Ο/AutfLβ(C1β), we get that in each of the above four spaces, the Aut(C1β)-orbits coincide with the Aut(C2β)-orbits, so the minimal Aut(C1β)-subflows coincide with the minimal Aut(C2β)-subflows. Hence, EL(IL,C1ββ):=cl(Aut(C1β)idIL,C1βββ)=cl(Aut(C2β)idIL,C1βββ). If the minimal left ideals in EL1β are of bounded size, then (by Proposition 5.5) they are contained in IL,C1βSXβ(C1β)β, so they are naturally minimal Aut(C2β)-flows, and for every minimal left ideal M1β of EL1β the restriction isomorphism F1β:M1ββIm(F1β)βEL(IL,C1ββ) of Aut(C1β)-flows (see Proposition 5.11(iv)) is also an isomorphism of Aut(C2β)-flows.
Let now Ο12β denote the homeomorphism from IL,C1ββ to IL,C2ββ. It induces a homeomorphism Ο12β²β:IL,C1βIL,C1ββββIL,C2βIL,C2βββ.
Remark 5.12*.*
i) Ο12β and Ο12β²β are both isomorphisms of Aut(C2β)-flows and Ο12β²β is an isomorphism of semigroups.
ii) Ο12β²β[EL(IL,C1ββ)]=EL(IL,C2ββ).
iii) Ο12β²β maps the collection of all minimal left ideals in EL(IL,C1ββ) bijectively onto the collection of all minimal left ideals in EL(IL,C2ββ).
Proof.
Point (i) follows from the definition of the action of Aut(C2β) given in the above discussion. Point (ii) follows from (i), namely: Ο12β²β[EL(IL,C1ββ)]=Ο12β²β[cl(Aut(C1β)idIL,C1βββ)]=Ο12β²β[cl(Aut(C2β)idIL,C1βββ)]=cl(Aut(C2β)idIL,C2βββ)=EL(IL,C2ββ). Point (iii) follows from (i) and (ii).
β
From now on, let Ο12β²β be the restriction of the old Ο12β²β to EL(IL,C1ββ). So now Ο12β²β:EL(IL,C1ββ)βEL(IL,C2ββ) is an isomorphism of Aut(C2β)-flows and of semigroups.
The next proposition gives us Theorem 0.4(ii) with some additional information.
Proposition 5.13**.**
*Assume that a minimal left ideal of the Ellis semigroup of SXβ(C) is of bounded size. Then:
i) For every minimal left ideal M1β in EL1β there exists a minimal left ideal M2β in EL2β which is isomorphic to M1β as a semigroup.
ii) For every minimal left ideal M2β in EL2β there exists a minimal left ideal M1β in EL1β which is isomorphic to M2β as a semigroup.
iii) All minimal left ideals in EL1β are naturally pairwise isomorphic minimal Aut(C2β)-flows and are isomorphic as Aut(C2β)-flows to the minimal Aut(C2β)-subflows of EL2β.*
Proof.
First of all, by assumption and Corollary 5.9, the minimal left ideals in EL1β and the minimal left ideals in EL2β are all of bounded size.
(i) Let M1β be a minimal left ideal of EL1β.
By Proposition 5.11, the restriction maps FΛ1β:EL1ββEL(IL,C1ββ) and FΛ2β:EL2ββEL(IL,C2ββ) are semigroup epimorphisms. Moreover, F1β:=FΛ1ββ£M1ββ:M1ββIm(F1β) is a semigroup isomorphism and Im(F1β) is a minimal left ideal of EL(IL,C1ββ). By the above discussion, Ο12β²β:EL(IL,C1ββ)βEL(IL,C2ββ) is an isomorphism, so Ο12β²β[Im(F1β)] is a minimal left ideal of EL(IL,C2ββ). Then FΛ2β1β[Ο12β²β[Im(F1β)]] is a left ideal of EL2β, which contains some minimal left ideal M2β. Then, by the minimality of Ο12β²β[Im(F1β)], we get that FΛ2β[M2β]=Ο12β²β[Im(F1β)], and, by Proposition 5.11, F2β:=FΛ2ββ£M2ββ:M2ββΟ12β²β[Im(F1β)] is a semigroup isomorphism. Therefore, F2β1ββΟ12β²ββ£Im(F1β)ββF1β:M1ββM2β is a semigroup isomorphism, and we are done.
(ii) This can be shown analogously to (i), but βgoing in the opposite directionβ.
(iii) All minimal left ideals of EL1β are pairwise isomorphic as Aut(C1β)-flows. So, by the description of the natural Aut(C2β)-flow structure on these ideals, we get that they are also isomorphic as Aut(C2β)-flows and that they are minimal Aut(C2β)-flows. Of course, all minimal left ideals (equivalently, minimal subflows) of EL2β are also isomorphic as Aut(C2β)-flows. Then, we apply the proof of (i), noticing that the discussion preceding Proposition 5.13 implies that F1β, F2β, and Ο12β²ββ£Im(F1β)β are all Aut(C2β)-flow isomorphisms.
β
Corollary 5.9 and Proposition 5.13 imply Theorem 0.4. By virtue of Proposition 2.4, we get the obvious counterpart of Theorem 0.4 for both SΞ±Λβ(C) and ScΛβ(C) in place of SXβ(C), which answers Question 0.3. An explicit bound on the size of the minimal left ideals is provided in Proposition 5.5 and Corollary 5.10.
6. The stable case
In the context of topological dynamics for definable groups, when the theory in question is stable, all notions (such as the Ellis semigroup, minimal ideals, the Ellis group) are easy to describe and coincide with well-known and important model-theoretic notions. This was a starting point to generalize some phenomena to wider classes of theories.
In this section, we are working in the Ellis semigroup of the
Aut(C)-flow ScΛβ(C), where C is a monster model of a
stable theory and cΛ is an enumeration of C, and we
give very easy descriptions of the minimal left ideals and of the
Ellis group. The reason why we work with ScΛβ(C) is that by
the appendix of [10] we know that stability is equivalent to the existence of a left-continuous semigroup operation β on ScΛβ(C) extending the action of Aut(C) (i.e. Ο(tp(cΛ/C))βq=Ο(q) for ΟβAut(C)), which implies that ScΛβ(C)β
EL(ScΛβ(C)) via the map pβ¦lpβ, where lpβ:ScΛβ(C)βScΛβ(C) is given by lpβ(q)=pβq. For a short tuple Ξ±Λ in place of cΛ this is not true (e.g. in the theory of an infinite set with equality, taking Ξ± to be a single element, we have that β£SΞ±β(C)β£=β£S1β(C)β£=β£Cβ£ while EL(S1β(C)) has size at least β£Aut(C)β£=2β£Cβ£). However, at the end of this section, as corollaries from our observations made for cΛ, we describe what happens in EL(SΞ±Λβ(C)).
From now on, in this section, T is a stable theory.
For any (short or long) tuple aΛ of elements of C and a (small or large) set of parameters B let
[TABLE]
NFaΛβ(C) is clearly a subflow of SaΛβ(C). Note that
NFaΛβ(acleq(β
))=SaΛβ(acleq(β
))
is also an Aut(C)-flow which is transitive. Let Ο΅Λ
denote an enumeration of acleq(β
). Then NFΟ΅Λβ(acleq(β
))=SΟ΅Λβ(acleq(β
)) is an Aut(C)-flow, and note
that SΟ΅Λβ(acleq(β
)) can be naturally
identified with the profinite group Aut(acleq(β
)) of
all elementary permutations of acleq(β
). Notice that
NFcΛβ(C) is closed under β.
The key basic consequences of stability that we will be using are:
each type over an algebraically closed set (e.g. a model) is stationary, i.e. it has a unique non-forking extension to any given superset,
NFaΛβ(C) is a transitive Aut(C)-flow,
the Aut(C)-orbit of each type in SaΛβ(C)βNFaΛβ(C) is of unbounded size.
Recall that AutfShβ(C):=Aut(C/acleq(β
)), EShβ is the orbit equivalence relation of AutfShβ(C) on any given product of sorts, and GalShβ(T):=Aut(C)/AutfShβ(C)β
Aut(acleq(β
)). In the stable context, AutfShβ(C)=AutfKPβ(C)=AutfLβ(C), so the corresponding orbit equivalence relations coincide, and the corresponding Galois groups coincide, too.
Remark 6.1*.*
Let aΛ be an enumeration of a (small or large) algebraically closed subset of C (e.g. aΛ=cΛ or aΛ=Ο΅Λ).
i) The restriction map r:NFcΛβ(C)βNFaΛβ(C) is a flow isomorphism. In particular, r induces a unique left-continuous semigroup operation (also denoted by β) on NFaΛβ(C)
such that whenever (Οiβ) is a net in Aut(C) satisfying limΟiβ(tp(aΛ/C))=pβNFaΛβ(C) and qβNFaΛβ(C), then pβq=limΟiβ(q).
ii) The restriction map rβ
β:NFaΛβ(C)βSaΛβ(acleq(β
)) is a flow isomorphism, which induces a unique left-continuous semigroup operation on SaΛβ(acleq(β
)) (still denoted by β) which coincides with the action of Aut(C) (i.e. Ο(tp(aΛ/acleq(β
)))βq=Ο(q) for ΟβAut(C) and qβSaΛβ(acleq(β
))).
iii) The restriction map R:NFaΛβ(C)βSΟ΅Λβ(acleq(β
))=Aut(acleq(β
)) is an isomorphism of Aut(C)-flows and of semigroups (where we take the obvious group structure on Aut(acleq(β
)). Thus, NFaΛβ(C) is a group.
Proof.
i) In the first part, only injectivity of r requires an
explanation. So consider p,qβNFcΛβ(C) such that
r(p)=r(q). Let Cβ²β»C be a bigger monster model in which we
will take all realizations. Choose aΛβ² realizing r(p)=r(q) and
extend it to cΛβ²β¨p and cΛβ²β²β¨q. Since cΛβ²β‘cΛβ²β², we get cΛβ²β‘aΛβ²βcΛβ²β². Since
acleq(β
)βaΛβ² we get that cΛβ²β‘acleq(β
)βcΛβ²β². Both p and q do not fork
over β
, hence p=q.
It is clear that r and the original semigroup operation on NFcΛβ(C) induce a left-continuous semigroup operation on NFaΛβ(C) such that for any net (Οjβ) in Aut(C) satisfying
limΟjβ(tp(cΛ/C))=pβ²βNFcΛβ(C) and any qβNFaΛβ(C) we have r(pβ²)βq=limΟjβ(q). Since each such a
net (Οjβ) satisfies limΟjβ(tp(aΛ/C))=r(pβ²), it is
enough to prove that for any net (Οiβ) in Aut(C) such that
limΟiβ(tp(aΛ/C))=pβNFaΛβ(C) and any qβNFaΛβ(C), the limit limΟiβ(q) exists. This in turn
follows since q is definable over
acleq(β
)βaΛ: if aΛ0ββacleq(β
) and Ο(yΛβ,aΛ0β) defines
qβ£Οβ, then (limΟiβ(q))β£Οβ is defined by
Ο(yΛβ,aΛ1β), where aΛ1β=limΟiβ(aΛ0β).
ii) The fact that rβ
β is a flow isomorphism is immediate from stationarity of the types over algebraically closed sets.
For the other part, let ΟβAut(C) and qβSaΛβ(acleq(β
)).
Take p0ββNFaΛβ(C) extending tp(aΛ/acleq(β
)). Since the Aut(C)-orbit of tp(aΛ/C) is dense in SaΛβ(C), we can find a net (Οiβ) in Aut(C) such that limΟiβ(tp(aΛ/C))=Ο(p0β). Then limΟiββ£acleq(β
)β=Οβ£acleq(β
)β. Also, Ο(tp(aΛ/acleq(β
))=rβ
β(Ο(p0β)).
Hence, by (i),
[TABLE]
iii) The fact that R is a flow isomorphism follows from (i) and (ii). To see that R is a semigroup isomorphism, one should check that the natural identification of SΟ΅Λβ(acleq(β
)) with Aut(acleq(β
)) is a homomorphism, where SΟ΅Λβ(acleq(β
)) is equipped with the semigroup operation from (ii). But this is obvious by (ii).
β
Proposition 6.2**.**
NFcΛβ(C)* is the unique minimal left ideal in ScΛβ(C), and β£NFcΛβ(C)β£β€2β£Tβ£.*
Proof.
Minimality of NFcΛβ(C) is clear, as NFcΛβ(C) is a transitive Aut(C)-flow.
By Remark 6.1, β£NFcΛβ(C)β£=β£SΟ΅Λβ(acleq(β
))β£β€2β£Tβ£ is bounded. Hence, all minimal left ideals are of bounded size.
On the other hand, stability implies that the Aut(C)-orbit of any pβScΛβ(C)βNFcΛβ(C) is unbounded. This shows uniqueness of the minimal left ideal.
β
So let M:=NFcΛβ(C) be the unique minimal left ideal of ScΛβ(C). Let uβM be an idempotent. By Remark 6.1(iii), M is a group, so M=uM and u is the unique idempotent in M. Moreover, the restriction map R:MβAut(acleq(β
)) is a group isomorphism, which explicitly shows absoluteness of the Ellis group of the flow ScΛβ(C).
Under the natural identification of EL(ScΛβ(C)) with ScΛβ(C) described in the second paragraph of this section, the semigroup epimorphism f^β:EL(ScΛβ(C))βGalLβ(T) from [10] (recalled in Section 1) is given by f^β(p)=Ο/AutfShβ(Cβ²) for any ΟβAut(Cβ²) such that Ο(cΛ)β¨pβScΛβ(C), where Cβ²β»C is a bigger monster model.
As was recalled in Section 1, f:=f^ββ£uMβ is a group epimorphism onto GalShβ(T). Using the natural identification of GalShβ(T) with Aut(acleq(β
)), one gets that R=f^ββ£Mβ, so we have
Corollary 6.3**.**
f:uMβGalShβ(T)* is a group isomorphism.*
From Remark 6.1(iii), we get
Corollary 6.4**.**
The unique idempotent uβM is the unique global non-forking extension of tp(cΛ/acleq(β
)).
Now we give a description of the group operation β on M=NFcΛβ(C).
Proposition 6.5**.**
Let p,q,rβNFcΛβ(C). Take any cΛβ²β¨q. Then pβq=r if and only if there exists ΟβAut(Cβ²) such that Ο(cΛ)β¨p and Ο(cΛβ²)β¨r.
Proof.
(β) There is a net (Οiβ) in Aut(C) such that limΟiβ=p, which formally means that limΟiβ(tp(cΛ/C))=p. By the left continuity of β, we get limΟiβ(q)=r. Thus, an easy compactness argument yields the desired Ο. (Note that this implication does not use the assumption that p,q,rβNFcΛβ(C).)
(β) This follows from Remark 6.1(iii), but we give a direct computation. Take Ο satisfying the right hand side. By (β), we can find Οβ²βAut(Cβ²) such that Οβ²(cΛ)β¨p and Οβ²(cΛβ²)β¨pβq. Since Ο(cΛ)β¨p, we get that Οβ£acleq(β
)β=Οβ²β£acleq(β
)β. Hence, rβ£acleq(β
)β=tp(Ο(cΛβ²)/acleq(β
))=Ο(tp(cΛβ²/acleq(β
)))=Οβ²(tp(cΛβ²/acleq(β
)))=tp(Οβ²(cΛβ²)/acleq(β
))=pβqβ£acleq(β
)β. This implies that r=pβq, because r,pβqβNFcΛβ(C).
β
As was recalled in the second paragraph of this section, there is a
semigroup isomorphism l:ScΛβ(C)βEL(ScΛβ(C))
given by l(p)=lpβ. By Proposition 6.2, l[M] is the unique minimal left ideal of EL(ScΛβ(C)) and it is of bounded size. In stable theories, Lascar invariant global
types are the types that do not fork over β
. Hence, by
Proposition 5.5, for every Ξ·βl[M], Im(Ξ·)βNFcΛβ(C), but since M=NFcΛβ(C) is a group and lpβ(q)=pβq, we easily conclude that Im(Ξ·)=NFcΛβ(C).
Now, consider a short tuple Ξ±Λ. By the above conclusions and Remark 2.1, we get
Corollary 6.6**.**
There is a unique minimal left ideal MΞ±Λβ in EL(SΞ±Λβ(C)), it coincides with the Ellis group, and for every Ξ·βMΞ±Λβ,
Im(Ξ·)=NFΞ±Λβ(C). The size of MΞ±Λβ is bounded by 2β£Tβ£.
Corollary 6.7**.**
If Ξ±Λ enumerates an algebraically closed set, then the unique minimal left ideal MΞ±Λβ in EL(SΞ±Λβ(C)) is isomorphic as a group with NFΞ±Λβ(C) (with the semigroup structure provided by Remark 6.1(i)) which is further isomorphic with Aut(acleq(β
)).
Proof.
By Remark 6.1 (i) and (iii), NFΞ±Λβ(C) is a group isomorphic with Aut(acleq(β
)) and the assignment pβ¦lpβ (where lpβ(q)=pβq) yields an isomorphism from NFΞ±Λβ(C) to EL(NFΞ±Λβ(C)).
Since NFΞ±Λβ(C) coincides with the set of Lascar invariant types in SΞ±Λβ(C), by the previous corollary together with Proposition 5.11,
we get that the restriction of the domains from SΞ±Λβ(C) to NFΞ±Λβ(C) yields a monomorphism from MΞ±ΛββNFΞ±Λβ(C)SΞ±Λβ(C) to EL(NFΞ±Λβ(C))βNFΞ±Λβ(C)NFΞ±Λβ(C) whose image is a minimal left ideal in EL(NFΞ±Λβ(C)) and so coincides with EL(NFΞ±Λβ(C)) (as EL(NFΞ±Λβ(C)) is a group by the first paragraph of this proof).
β
7. The NIP case
Throughout this section, we assume that C is a monster model of a theory T with NIP. Let ΞΊ be the degree of saturation of C. As usual, cΛ is an enumeration of C, and Ξ±Λ is a short tuple in C. Let S be an arbitrary product of sorts (with repetitions allowed), and X be a β
-type-definable subset of S. Let M be a minimal left ideal of EL(SXβ(C)).
In the first subsection, after giving some characterizations of when the minimal left ideals in EL(SXβ(C)) are of bounded size, we prove the main result of the first subsection which yields several characterizations (in various terms) of when the minimal left ideals of EL(ScΛβ(C)) are of bounded size. We also make some observations and state questions concerning boundedness of the minimal left ideals of EL(SΞ±Λβ(C)).
In the second subsection, we give a better bound (than the one from Corollaries 4.4 and 4.5) on the size of the Ellis group of the flow SXβ(C), and, as a consequence β of the flows ScΛβ(C) and SΞ±Λβ(C). The main point is that instead of the set R obtained in Section 4 via contents, under the NIP assumption we can just use types invariant over a model.
In the last subsection, we adapt the proof of [3, Theorem 5.7] to show Theorem 0.7. We also find a counterpart of the epimorphism f described in Section 1 which goes from the Ellis group of SΞ±Λβ(C) to a certain new Galois group introduced in [4], and we
give an example showing that the obvious counterpart of Theorem 0.7 does not hold for this new epimorphism.
7.1. Characterizations of boundedness of minimal left ideals
Recall that all the time we assume NIP.
Fact 7.1**.**
*Let pβSXβ(C).
Then, the following conditions are equivalent.
i) The Aut(C)-orbit of p is of bounded size.
ii) p does not fork over β
.
iii) p is Kim-Pillay invariant (i.e. invariant under AutfKPβ(C)).
iv) p is Lascar invariant.*
Proof.
(i) β (ii). Suppose p forks over β
. Then there is Ο(xΛ,aΛ)βp which divides over β
. So there is a tuple (aiβ)i<ΞΊβ in C which is indiscernible over β
and such that the sequence β¨Ο(xΛ,aΛiβ):i<ΞΊβ© is k-inconsistent for some k<Ο. This implies that the orbit Aut(C)p is of size at least ΞΊ which is unbounded.
The implication (ii) β (iii) is Proposition 2.11 of [7]. The implications (iii) β (iv) and (iv) β (i) are obvious.
β
An immediate corollary of this fact and Proposition 5.5 is
Corollary 7.2**.**
*The following conditions are equivalent.
i) M is of bounded size.
ii) For every Ξ·βM the image Im(Ξ·) consists of types which do not fork over β
.
iii) For some Ξ·βEL the image Im(Ξ·) consists of types which do not fork over β
.*
Proposition 7.3**.**
*The following conditions are equivalent.
i) M is of bounded size.
ii) For every natural number n, for every types q1β,β¦,qnββSXβ(C) and for every formulas Ο1β(xΛ,aΛ1β),β¦,Οnβ(xΛ,aΛnβ) (where xΛ corresponds to S) which fork over β
, there exists ΟβAut(C) such that Ο(Οiβ(xΛ,aΛiβ))β/qiβ for all i=1,β¦,n.
iii) The same condition as (ii) but with βforkingβ replaced by βdividingβ.
iv) For every natural number n, for every type qβSXβ(C) and for every formulas Ο1β(xΛ,aΛ1β),β¦,Οnβ(xΛ,aΛnβ) (where xΛ corresponds to S) which fork over β
, there exists ΟβAut(C) such that Ο(Οiβ(xΛ,aΛiβ))β/q for all i=1,β¦,n.
v) The same condition as (iv) but with βforkingβ replaced by βdividingβ.
vi) For every qβSXβ(C) the closure cl(Aut(C)q) of the Aut(C)-orbit of q contains a type which does not fork over β
.*
Proof.
(i) β (ii). Consider any types q1β,β¦,qnββSXβ(C) and formulas Ο1β(xΛ,aΛ1β),β¦,Οnβ(xΛ,aΛnβ) which fork over β
. Take Ξ·βM. By Corollary 7.2, Ξ·(qiβ) does not fork over β
, and so Β¬Οiβ(xΛ,aΛiβ)βΞ·(qiβ), for all i=1,β¦,n. Since Ξ· is approximated by automorphisms, there is Οβ²βAut(C) such that Β¬Οiβ(xΛ,aΛiβ)βΟβ²(qiβ) for i=1,β¦,n. Thus, Ο:=Οβ²β1 does the job.
The implications (ii) β (iii), (ii) β (iv), (iii) β (v), (iv) β (v) are obvious.
(v) β (vi). This follows immediately by taking the limit of a convergent subnet of the net of inverses of the automorphisms which we get for all possible finite sequences of formulas as in (v). (Here, the partial directed order on finite sequences of dividing formulas is given by being a subsequence.)
(vi) β (i). Consider any Ξ·βM. By Corollary 7.2, it is enough to show that Ξ·(q) does not fork over β
for all qβSXβ(C). So take any qβSXβ(C).
Let Οqβ:MβSXβ(C) be given by
Οqβ(ΞΎ)=ΞΎ(q). This is a homomorphism of Aut(C)-flows, so
Im(Οqβ) is a minimal subflow of SXβ(C), hence
Im(Οqβ)=cl(Aut(C)p) for every pβIm(Οqβ). By (vi), there
is pβIm(Οqβ) that does not fork over β
. By invariance
and closedness of the collection of non-forking types, every type in
Im(Οqβ) does not fork over β
, in particular
Οqβ(Ξ·)=Ξ·(q) does not fork over β
.
β
We will say that a formula Ο(xΛ,aΛ) is weakly invariant in X if the collection of formulas {Ο(Ο(xΛ,aΛ)):ΟβAut(C)}βͺX is consistent (where X is treated as a partial type over β
). Equivalently, this means that the collection of formulas {Ο(Ο(xΛ,aΛ)):ΟβAut(C)} extends to a type in SXβ(C).
Corollary 7.4**.**
M* is of bounded size if and only if each formula which is weakly invariant in X does not fork over β
.*
Proof.
(β) This implication does not require NIP, although the
NIP assumption greatly simplifies the proof. Suppose for a
contradiction that Ο(xΛ):={Ο(Ο(xΛ,aΛ)):ΟβAut(C)} extends to a type pβSXβ(C), but
Ο(xΛ,aΛ) forks over β
. Note that for
every ΟβAut(C), Ο(xΛ)βΟ(p) and
Ο(xΛ) forks over β
. Using NIP, by Proposition
7.3, we get
ΟβAut(C) with Ο(Ο(xΛ,aΛ))ξ βp,
a contradiction. Without NIP we proceed as follows.
Let [Ο(xΛ)]={qβSXβ(C):Ο(xΛ)βq}.
Let Οpβ:EL(SXβ(C))βSXβ(C) be given by
Οpβ(Ξ·)=Ξ·(p). This is a homomorphism of
Aut(C)-flows. Therefore, Im(Οpβ)=cl(Aut(C)p)β[Ο(xΛ)]. So for every Ξ·βEL(SXβ(C)), Ξ·(p) forks
over β
. By (1)β(2) in Fact 7.1 (which does not use NIP), the orbit
Aut(C)Ξ·(p) is of unbounded size which implies that every
Aut(C)-orbit in EL(SXβ(C)) is of unbounded size, and so is
M, a contradiction.
(β) We check that item (iv) from Proposition 7.3 holds. Since in (iv) we are talking about forking (and not about dividing), it is enough to show that for any qβSXβ(C) and a formula Ο(xΛ,aΛ) which forks over β
there is ΟβAut(C) such that Ο(Ο(xΛ,aΛ))β/q. But this is clear, because by assumption, the fact that Ο(xΛ,aΛ) forks over β
implies that {Ο(Ο(xΛ,aΛ)):ΟβAut(C)}βͺX is inconsistent.
β
Note that each formula Ο(xΛ,aΛ) weakly invariant in
X does not divide over β
, and even Ο(xΛ,aΛ)βͺX does not divide over β
. Therefore, by the last
corollary, if for each formula Ο(xΛ,aΛ) such that
Ο(xΛ,aΛ)βͺX does not divide over β
we
have that Ο(xΛ,aΛ)βͺX does not fork over
β
(in such a case, we will say that forking equals
dividing on X), then M is of bounded size. Does the
converse hold?
Question 7.5**.**
Is it true that M is of bounded size if and only if forking equals dividing on X?
Now, we recall a few notions. A subset D of a point-transitive G-flow Y (i.e. Y=cl(Gy) for some yβY) is said to be:
generic (or syndetic) if finitely many translates of D by elements of G cover Y,
weakly generic if there is a non-generic FβY such that DβͺF is generic.
The first notion is classical in topological dynamics and in model
theory. The second one was introduced by the second author as a
substitute for the notion of a generic set in the unstable
context. Recall that pβY is called almost periodic if it
belongs to a minimal subflow of Y; it is [weakly] generic if
every open subset of Y containing p is [weakly] generic. The
second author proved that the set of all weakly generic points is the
closure of the set of all almost periodic points, and he suggested
that in the context of a group G definable in a model M acting on
SGβ(M), the notion of a weak generic type may be the right
substitute of the notion of a generic type in an unstable context [12].
Let us return to our context. From now on, we will consider the case where X=tp(cΛ/β
) or X=tp(Ξ±Λ/β
). Then SXβ(C) is a point-transitive Aut(C)-flow (it is an Aut(C)-ambit with the distinguished point tp(cΛ/β
) or tp(Ξ±Λ/β
)). We adapt the above general terminology to our context and say that a formula Ο(xΛ,aΛ) is [weakly] generic in X if [Ο(xΛ,aΛ)]β©SXβ(C) is a [weakly] generic subset of the Aut(C)-flow SXβ(C). Note that a formula Ο(xΛ,aΛ) is weakly invariant in X (in our earlier terminology) if and only if Β¬Ο(xΛ,aΛ) is not generic in X. A type pβSXβ(C) is [weakly] generic if it is so as an element of the Aut(C)-flow SXβ(C), which is equivalent to saying that every formula in p is [weakly] generic in X. These notions were considered in this context already in [15] under the name of β[weakly] c-freeβ instead of β[weakly] genericβ. Similarly, p is said to be almost periodic if it is so as an element of the flow.
Recall that a set A is called an extension base if every type pβS(A) does not fork over A. This is equivalent to saying that each pβS(A) has a non-forking extension to any given superset of A.
The following facts come from [2].
Fact 7.6**.**
*Recall that we assume NIP.
i) Each model is an extension base (also without NIP).
ii) A set A is an extension base if and only if forking equals dividing over A (i.e. a formula Ο(yΛβ,aΛ) divides over A iff it
forks over A).*
Now, we will prove our main characterization result in the case of X=tp(cΛ/β
). In particular, it contains Proposition 0.6.
Theorem 7.7**.**
*Let X=tp(cΛ/β
) (so M is a minimal left ideal of EL(ScΛβ(C)). Then the following conditions are equivalent.
i) β
is an extension base.
ii) Forking equals dividing on X.
iii) The weakly generic formulas in X do not fork over β
.
iv) The almost periodic types in ScΛβ(C) have orbits of bounded size.
v) M is of bounded size.
vi) There is a type in ScΛβ(C) whose Aut(C)-orbit is of bounded size; equivalently, there is a type in ScΛβ(C) which does not fork over β
; equivalently, tp(cΛ/β
) does not fork over β
.*
Proof.
(i) β (ii). By Fact 7.6(ii), point (i) implies that forking equals dividing over β
, let alone forking equals dividing on X.
(ii) β (iii).
This is essentially [15, Lemma 5.9].
Consider Ο(xΛ,aΛ) which is weakly generic in X. Take a formula Ο(xΛ,bΛ) which is non-generic in X and such that Ο(xΛ,aΛ)β¨Ο(xΛ,bΛ) is generic in X. Then there are Ο1β,β¦,ΟnββAut(C) such that
for Οβ²(xΛ,aΛβ²):=Ο1β(Ο(xΛ,aΛ))β¨β―β¨Οnβ(Ο(xΛ,aΛ)) and Οβ²(xΛ,bΛβ²):=Ο1β(Ο(xΛ,bΛ))β¨β―β¨Οnβ(Ο(xΛ,bΛ)) one has
[TABLE]
Suppose for a contradiction that Ο(xΛ,aΛ) forks over β
. Then Οβ²(xΛ,aΛβ²) also forks over β
.
By (ii), Οβ²(xΛ,aΛβ²)βͺX divides over β
, so there are Ο1β,β¦,ΟmββAut(C) such that
[TABLE]
Since ([Οiβ(Οβ²(xΛ,aΛβ²))]βͺ[Οiβ[Οβ²(xΛ,bΛβ²)])β©SXβ(C)=SXβ(C) for all i=1,β¦,m, we get that SXβ(C)ββiβ€mβ[Οiβ(Οβ²(xΛ,bΛβ²))]ββiβ€mβ[Οiβ(Οβ²(xΛ,aΛβ²)]β©SXβ(C)=β
. Hence, Ο(xΛ,bΛ) is generic in X, a contradiction.
(iii) β (iv). Each almost periodic type is weakly generic
[12, Corollary 1.8], hence it does not fork over β
by (iii), and so its orbit is of bounded size by Fact 7.1.
(iv) β (v). Since for every qβSXβ(C) the function Οqβ:MβSXβ(C) defined by Οqβ(Ξ·):=Ξ·(q) is a homomorphism of Aut(C)-flows, we see that Im(Οqβ) is a minimal subflow of SXβ(C). Thus, we get that for every Ξ·βM, Im(Ξ·) consists of almost periodic types, which by (iv) implies that each type in Im(Ξ·) has bounded orbit. Hence, by Fact 7.1, for every Ξ·βM, Im(Ξ·) consists of types which do not fork over β
, so M is of bounded size by Corollary 7.2.
(v) β (vi). It is enough to take any Ξ·βM and a type in the image of Ξ·. The orbit of this type is of bounded size by Proposition 5.5.
(vi) β (i). Since cΛ enumerates the whole monster model, the fact that some type in ScΛβ(C) does not fork over β
implies that every type in S(β
) (in arbitrary variables) does not fork over β
, i.e. β
is an extension base.
β
Alternatively, one could prove (i) β (v) directly.
Namely, by (i) β (ii), if β
is an extension base, then forking equals dividing on X. By the paragraph following Corollary 7.4, we conclude that M is of bounded size.
Question 7.5 is quite interesting.
Note that Theorem 7.7 shows that the answer is positive in the case when X=tp(cΛ/β
).
By the above theorem and [7, Corollary 2.10], we immediately get
Corollary 7.8**.**
If M is of bounded size, then the theory T is G-compact.
The proof of Theorem 7.7 yields the next proposition.
Proposition 7.9**.**
*Let X=tp(Ξ±Λ/β
) (so M is a minimal left ideal of EL(SΞ±Λβ(C)). Consider the following conditions.
i) β
is an extension base.
ii) Forking equals dividing on X.
iii) The weakly generic formulas in X do not fork over β
.
iv) The almost periodic types in SΞ±Λβ(C) have orbits of bounded size.
v) M is of bounded size.
vi) tp(Ξ±Λ/β
) does not fork over β
.
Then (i) β (ii) β (iii) β (iv) β (v) β (vi).*
Question 7.10**.**
Are Conditions (ii) - (v) in the above proposition equivalent?
To see that (v) does not imply (i), even under NIP, take a 2-sorted structure with sorts S1β and S2β such that there is no structure on S1β, there are no interactions between S1β and S2β, and the structure on S2β is such that β
is not an extension base. Then, taking any Ξ±βS1β, we get that
the minimal left ideal of EL(SΞ±β(C)) is trivial, but (i) does not hold.
It is more delicate to build an example showing that (vi) does not imply (v). This will be dealt with in the appendix.
7.2. A better bound on the size of the Ellis group
Under the NIP assumption, we will give a better bound on the size of the Ellis group of the flow SXβ(C) than the one in Corollaries 4.4 and 4.5.
Instead of contents and the family R of types used in Section 4, we will show that one can use the family of global types invariant over a given small model M. The families of global types invariant over M considered over monster models C1ββ»C2β can be also used instead of the families RC1ββ and RC2ββ obtained in Corollary 4.14 to get absoluteness of the Ellis group (namely, using these families, one can still show Lemmas 4.15, 4.16, 4.17, and 4.18, to get Corollary 4.19; the further material on Ο-topologies also goes through).
So fix a small model MβΊC, and let IMβ be the family of all types in SXβ(C) which are invariant over M. Note that by Fact 7.1, IMβ coincides with the set of all types in SXβ(C) which do not fork over M.
Recall that M denotes a minimal left ideal of EL(SXβ(C)).
The key observation is the following.
Proposition 7.11**.**
There exists an idempotent uβM such that Im(u)βIMβ.
Proof.
By Lemma 4.3, it is enough to show that there exists Ξ·βEL(SXβ(C)) such that Im(Ξ·)βIMβ. For a type pβSXβ(C) and a formula Ο(xΛ,aΛ) which forks over M, let
[TABLE]
It is is enough to show that the intersection of all possible sets
Xp,Οβ is non-empty, as then any element Ξ· in this
intersection will do the job. By compactness of EL(SXβ(C)), it
remains to show that the family of sets of the form Xp,Οβ
has the finite intersection property. So consider any types
p1β,β¦,pnββSXβ(C) and any formulas Ο1β(xΛ,aΛ1β),β¦,Οnβ(xΛ,aΛnβ) which all fork over M, and
let Ο(xΛ,aΛ) be the disjunction of them.
By Fact 7.6,
Ο divides over M, so choose an M-indiscernible sequence
(bΛjβ)j<Οβ witnessing this. For each i=1,β¦,n there
are at most finitely many j with Ο(xΛ,bΛjβ)βpiβ. So chosse j such that Ο(xΛ,bΛjβ)ξ βpiβ
for i=1,β¦,n. Let ΟβAut(C) map bΛjβ to aΛ. Then Ο, regarded as an element of EL(SXβ(C)), belongs to
Xp1β,Ο1βββ©β―β©XpnβΟnββ.
β
Recall the following fact (see e.g. [7, Lemma 2.5]).
Fact 7.12**.**
If T is NIP and AβC, then there are at most 2β£Tβ£+β£xΛβ£+β£Aβ£ types in SxΛβ(C) which are invariant over A.
As usual, lSβ denotes the length of the product S.
Choose any idempotent u satisfying the conclusion of Proposition 7.11.
Corollary 7.13**.**
The function F:uMβIMIMββ given by F(h)=hβ£IMββ is a group isomorphism onto the image of F, so the size of the Ellis group of the flow SXβ(C) is bounded by β£IMββ£β£IMββ£. In particular, the size of the Ellis group is bounded by
22lSβ+β£Tβ£. In the case when lSββ€2β£Tβ£, this is bounded by βΆ3β(β£Tβ£), and when lSββ€β£Tβ£, the bound equals 22β£Tβ£.
Proof.
The first part follows from Proposition 7.11 and Lemma 4.2(3). For the second part, take M of cardinality at most β£Tβ£. Then β£IMββ£β€2lSβ+β£Tβ£ by Fact 7.12, and we finish using the first part.
β
By Corollary 2.2, Propositions 2.4 and 2.6, and Corollary 7.13, we get
Corollary 7.14**.**
The size of the Ellis group of the flow SXβ(C) is bounded by βΆ3β(β£Tβ£). The sizes of the Ellis groups of the flows ScΛβ(C) and SΞ±Λβ(C) are bounded by 22β£Tβ£.
As was mentioned at the beginning of this subsection, in the NIP case, one can also simplify the proof of absoluteness of the Ellis group from Section 4 by omitting technical lemmas 4.7, 4.8, 4.10 and Corollary 4.12, and then proceeding with RC1ββ and RC2ββ replaced by types invariant over M (and using Proposition 7.11). We leave the details as an exercise.
7.3. The Ellis group conjecture for groups of automorphisms
Recall that the Ellis group conjecture was formulated by the second
author in the case of a group G definable in a model M. It says
that a certain natural epimorphism Ξ¦ (more precisely, taking the
coset of a realization of the given type) from the Ellis group of the
flow SG,extβ(M) (of all external types over M) to
Gβ/GβM00β is an isomorphism, at least under some reasonable
assumptions. In general, this turned out to be false,
e.g. G:=SL2β(R) treated as a group definable in the field of reals
(so an NIP structure) is a counter-example [5]. Much more
counter-examples can be obtained via the obvious observation that
the epimorphism Ξ¦ factors through Gβ/GβM000β (see
[8]), namely we get a counter-example whenever GβM000βξ =GβM00β. On the other hand, [3, Theorem 5.7]
confirms the Ellis group conjecture for definably amenable groups definable in NIP theories.
Here, we study an analogous problem for the group Aut(C) in place
of the definable group G. Take the notation from Section
1, so M is a minimal left ideal of
EL:=EL(ScΛβ(C)), uβM is an idempotent, and
we have maps f^β:ELβGalLβ(T),Β f:uMβGalLβ(T) and h:GalLβ(T)βGalKPβ(T). We get the epimorphism F^:=hβf^β:ELβGalKPβ(T). Let F=F^β£uMβ:uMβGalKPβ(T). This is also an epimorphism.
A natural counterpart of the Ellis group conjecture in our context says that F is an isomorphism. It is clearly false for all non G-compact theories (as then h is not injective). But we will prove it under the assumption of boundedness of the minimal left ideals in EL, and this is the precise content of Theorem 0.7.
Theorem 0.7.
Assume NIP. If M is of bounded size, then F is an isomorphism.
Note that boundedness of the minimal left ideals in EL exactly corresponds to the definable amenability assumption for definable NIP groups, because both conditions are equivalent to the existence of
a type (in the type space appropriate for each of the two contexts) with bounded orbit
(see [3, Theorem 3.12] for the proof of this in the definable group case). In fact,
the first author has proved that boundedness of the minimal left ideals in EL is equivalent to the appropriate version of amenability of Aut(C) (which he calls relative definable amenability), but this belongs to a separate topic.
The proof of Theorem 0.7 will be an adaptation of [3, Theorem 5.7].
Before the proof, let us recall some notation. Suppose that pβS(C) is invariant over A. Then a sequence (aiβ)i<Ξ»β is called a Morley sequence in p over A if aiββ¨pβ£Aa<iββ for all i. Such a sequence is always indiscernible over A, and the type of a Morley sequence of length Ξ» in p over A does depend on the choice of this sequence and is denoted by p(Ξ»)β£Aβ.
Lemma 7.15** (Counterpart of Proposition 5.1 of [3]).**
Let Ο(xΛ,aΛ) be a formula, Ξ±Λ any (short or long) tuple in C, and let
pβSΞ±Λβ(C)
be a type whose Aut(C)-orbit is of bounded size. Put UΟβ:={Ο/AutfKPβ(C):Ο(xΛ,Ο(aΛ))βp}. Then UΟβ is constructible (namely, a Boolean combination of closed sets).
Proof.
Let S={bΛ:Ο(xΛ,bΛ)βp}, V={ΟβAut(C):Ο(aΛ)βS}, Ο:Aut(C)βAut(C)/AutfKPβ(C) be the quotient map, and Ο:Aut(C)βC be given by Ο(Ο)=Ο(aΛ). Then V=Οβ1[S] and UΟβ=Ο[V].
By assumption and Fact 7.1, p
is invariant under AutfKPβ(C) and does not fork over
β
. Choose a small model MβΊC and let dΛ realize
pβ²(Ο)β£Mβ, where pβ² is the restriction of p to the finitely many variables occurring in Ο(xΛ,aΛ); from now on, xΛ is this finite tuple of variables. Let
[TABLE]
By NIP and the argument from Proposition 2.6 of [7], there is N<Ο such that
[TABLE]
where Anβ is the collection of all tuples bΛ in C for which
[TABLE]
and Bnβ is the collection of all tuples bΛ in C for which
[TABLE]
We see that each Anβ and Bnβ is invariant under AutfKPβ(C), so Οβ1[Anβ] and Οβ1[Bnβ] are unions of AutfKPβ(C)-cosets, i.e. they are unions of cosets of ker(Ο). Therefore,
[TABLE]
[TABLE]
So it remains to check that Ο[Οβ1[Anβ]] and Ο[Οβ1[Bnβ]] are closed.
Using the observation that Οβ1[Anβ] and Οβ1[Bnβ] are unions of cosets of ker(Ο), we get
[TABLE]
[TABLE]
By [11, Lemma 4.10], [1, Fact 2.3(i)], and the fact that the topology on GalKPβ(T) is the quotient topology induced from GalLβ(T), since Anβ and Bnβ are type-definable, we conclude that Ο[Οβ1[Anβ]] and Ο[Οβ1[Bnβ]] are closed.
β
As usual, Cβ²β»C is a monster model with respect to C. Recall that F^:=hβf^β:ELβGalKPβ(T) is given by F^(Ξ·)=Οβ²/AutfKPβ(Cβ²), where Οβ²βAut(Cβ²) is such that Οβ²(cΛ)β¨Ξ·(tp(cΛ/C)). Enumerate ScΛβ(C) as β¨tp(cΛkβ/C):k<Ξ»β© for some cardinal Ξ» and some tuples cΛkββ‘cΛ, where cΛ0β=cΛ. Let qkβ=tp(cΛkβ/C) for k<Ξ». By [10, Remark 2.3], it is true that for any k<Ξ», F^(Ξ·)=Οβ²/AutfKPβ(Cβ²), where Οβ²βAut(Cβ²) is such that Οβ²(cΛkβ)β¨Ξ·(qkβ). Let Οkβ:ELβScΛβ(C) be given by Οkβ(Ξ·)=Ξ·(qkβ).
Lemma 7.16** (Counterpart of Theorem 5.3 of [3]).**
Assume that Ξ·βEL, pβScΛβ(C) and k<Ξ» are
such that Οkβ(Ξ·)=p and the Aut(C)-orbit of p is of
bounded size. Let C=cl(Aut(C)Ξ·)βEL, and
Ο(xΛ,aΛ) be a formula over C. Put
[TABLE]
Then EΟβ is closed with empty interior.
Proof.
Since F^ is continuous (by Remark 2.5 of [10]), we get that EΟβ is closed.
Replacing Ξ· by some element from the Aut(C)-orbit of Ξ·, we can assume that p=tp(Ο(cΛkβ)/C) for some ΟβAutfKPβ(Cβ²).
Then F^(Ξ·)=Ο/AutfKPβ(Cβ²)=id/AutfKPβ(Cβ²).
By assumption and Fact 7.1, p is AutfKPβ(C)-invariant, so we can extend it uniquely to an AutfKPβ(Cβ²)-invariant type pΛββScΛβ(Cβ²).
Let UΟβ be the set defined in Lemma 7.15
(but for the type pΛβ in place of p, and working in Cβ²). We will show that EΟβββUΟβ1β,
where βA is the topological border of A: βA=cl(A)β©cl(Ac). This will finish the proof, as the topological border of a constructible set always has empty interior.
Consider any gβEΟβ and an open neighborhood V of g. By the definition of EΟβ, there are q,qβ²βC such that F^(q)=F^(qβ²)=g and qβΟkβ1β[[Ο(xΛ,aΛ)]], qβ²βΟkβ1β[[Β¬Ο(xΛ,aΛ)]]. Then q and qβ² belong to the open set F^β1[V]. So there are Ο1β,Ο2ββAut(C) such that Ο1βΞ·βF^β1[V]β©Οkβ1β[[Ο(xΛ,aΛ)]] and Ο2βΞ·βF^β1[V]β©Οkβ1β[[Β¬Ο(xΛ,aΛ)]].
Take any ΟΛ1ββAut(Cβ²) extending Ο1β.
We have ΟΛ1βpΛββΟ1βp=Ο1βΟkβ(Ξ·)=Οkβ(Ο1βΞ·)βΟ(xΛ,aΛ), and so Ο(xΛ,ΟΛ1β1β(aΛ))βpΛβ, hence F^(Ο1β)=ΟΛ1β/AutfKPβ(Cβ²)βUΟβ1β. On the other hand, we clearly have F^(Ο1βΞ·)βV. Since F^ is a semigroup homomorphism and F^(Ξ·) is the neutral element, we conclude that
[TABLE]
Similarly (and using the fact that pΛβ is AutfKPβ(Cβ²)-invariant),
[TABLE]
As V was an arbitrary open neighborhood of g, we get that gββUΟβ1β.
β
Now, we have all the tools to prove Theorem 0.7 (which is a counterpart of Theorem 5.7 of [3]).
Proof of Theorem 0.7.
Recall that F=F^β£uMβ:uMβGalKPβ(T) is an epimorphism.
We need to show that ker(F)={u}. So take any Ξ·βker(F). It is enough to show that rΞ·=r for some rβM, because then Ξ·=(ur)β1urΞ·=(ur)β1ur=u. But this is equivalent to finding rβM such that Οkβ(rΞ·)=Οkβ(r) for all k<Ξ».
Let F be the filter of comeager subsets of
GalKPβ(T). Since GalKPβ(T) is compact Hausdorff, it is a
Baire space, so F is a proper filter. Let Fβ² be an ultrafilter extending F. For any gβGalKPβ(T) let rgββM be such that F^(rgβ)=g. Put
[TABLE]
Then rβM, and we will show that Οkβ(rΞ·)=Οkβ(r) for all k, which completes the proof.
Suppose for a contradiction that Οkβ(rΞ·)ξ =Οkβ(r) for some k. Then Ο(xΛ)βΟkβ(rΞ·) and Β¬Ο(xΛ)βΟkβ(r) for some formula Ο(xΛ) with parameters. Thus,
[TABLE]
Since Ξ·βM and M is of bounded size, so is the Aut(C)-orbit of Οkβ(Ξ·), and we can apply Lemma 7.16. To be consistent with the notation from this lemma, put C:=M=cl(Aut(C)Ξ·).
By Lemma 7.16, EΟcββFβ², so we can find gβPβ©EΟcβ.
Put S=F^β1[EΟcβ]. Then S is open in EL and rgββS. Moreover, rgβΞ· belongs to the open set Οkβ1β[[Ο(xΛ)]] and rgβ belongs to the open set Οkβ1β[[Β¬Ο(xΛ)]].
Since rgβ is approximated by automorphisms of C, rgβ=rgβu, and the semigroup operation in EL is left continuous, we get that there is ΟβAut(C) such that
[TABLE]
We conclude that
[TABLE]
But
[TABLE]
implies that F^(Ο)=F^(ΟΞ·)=F^(Οu)βEΟβ, which is a contradiction.
β
We have seen in Corollary 7.8 that boundedness of M implies that the theory T is G-compact. Note that this also follows immediately from Theorem 0.7 (i.e. the epimorphism h:GalLβ(T)βGalKPβ(T) from Section 1 is an isomorphism).
As was mentioned in the introduction, although by Section 4 we know that the Ellis group uM is always bounded
and absolute, taking any non G-compact theory, we get that F is not
injective (i.e. the counterpart of the Ellis group conjecture is false in general, even in NIP theories). Corollary 1.4 shows even more, namely that if T is non G-compact, then the epimorphism f:uMβGalLβ(T) is not an isomorphism. It is thus still interesting to look closer at the Ellis group uM β an essentially new model-theoretic invariant of an arbitrary theory.
Now, we take the opportunity and describe a natural counterpart of the epimorphisms f and F for a short tuple Ξ±Λ in place of cΛ. But first note that in general there is no chance to find an epimorphism from the Ellis group of SΞ±Λβ(C) to GalKPβ(T) for the same reason as (v) β (i) in Proposition 7.9: Take a 2-sorted structure with sorts S1β and S2β such that there is no structure on S1β, there are no interactions between S1β and S2β, and the structure on S2β is such that the corresponding GalKPβ is non-trivial. Then, taking any Ξ±βS1β, we get that the Ellis group of the flow SΞ±β(C) is trivial, whereas GalKPβ(T) is non-trivial, so there is no epimorphism. In order to resolve this problem, we need to use a certain localized version of the notion of Galois group proposed in [4].
Let p1β=tp(Ξ±Λ/β
). Following the notation from [4], we define GalLfix,1β(p1β) as the quotient of the group of all elementary permutations of p1β(C) by the subgroup AutfLfix,1β(p1β(C)) of all elementary permutations of p1β(C) fixing setwise the ELβ-class of each realization of p1β (but not necessarily of each tuple of realizations of p1β, and that is why we write superscript 1). GalLfix,1β(p1β) can be, of course, identified with the quotient of Aut(C) by the group AutfL,p1βfix,1β(C) of all automorphisms fixing setwise the ELβ-class of each realization of p1β. The group GalKPfix,1β(p1β) is defined analogously. It is easy that these groups do not depend on the choice of the monster model. We have the following diagram of natural epimorphisms.
In particular, on GalLfix,1β(p1β) and GalKPfix,1β(p1β) we can take the quotient topologies coming from the vertical maps, and it is easy to check that GalLfix,1β(p1β) is a quasi-compact topological group, GalKPfix,1β(p1β) is a compact topological group, h1β is a topological quotient mapping (i.e. a subset of GalKPfix,1β(p1β) is closed if and only if its preimage is closed),
but there is no reason why ker(h1β) should be the closure of the identity in GalLfix,1β(p1β) (it certainly contains it).
On can easily check that both groups GalLfix,1β(p1β) and GalKPfix,1β(p1β) do not depend on the choice of C as topological groups.
Now, we will define a counterpart of the map f^β recalled in Section 1. We enumerate SΞ±Λβ(C) as β¨tp(Ξ±Λkβ/C):k<Ξ»β© for some cardinal Ξ» and some tuples Ξ±Λkββ‘Ξ±Λ, where Ξ±Λ0β=Ξ±Λ. Let qkβ=tp(Ξ±Λkβ/C) for k<Ξ».
Let Cβ²β»C be a monster model with respect to C.
As in [10, Proposition 2.3], using compactness and the density of Aut(C) in EL(SΞ±Λβ(C)), we easily get
Remark 7.17*.*
For every Ξ·βEL(SΞ±Λβ(C)) there is Οβ²βAut(Cβ²) such that for all k<Ξ», Ξ·(tp(Ξ±Λkβ/C))=tp(Οβ²(Ξ±Λkβ)/C). More generally, given any sequence β¨Ξ²Λβkβ:kβIβ©, where I is any set of size bounded with respect to Cβ² and {tp(Ξ²Λβkβ/C):kβI}βSΞ±Λβ(C), for every Ξ·βEL(SΞ±Λβ(C)) there is Οβ²βAut(Cβ²) such that for all kβI, Ξ·(tp(Ξ²Λβkβ/C))=tp(Οβ²(Ξ²Λβkβ)/C).
By this remark and the definition of GalLfix,1β(p1β) (which here will be computed using Cβ²), one easily gets that f1β^β:EL(SΞ±Λβ(C))βGalLfix,1β(p1β) given by
[TABLE]
where Οβ²βAut(Cβ²) is such that Οβ²(Ξ±Λkβ)β¨Ξ·(qkβ) for all k<Ξ», is a well-defined function. (To see that the value f^β1β(Ξ·) does not depend on the choice of Οβ², one should use the obvious fact that among Ξ±Λkβ, k<Ξ», there are representatives of all ELβ-classes on p1β(Cβ²).) Note that the difference in comparison with f^β is that here we have to use all types from SΞ±Λβ(C) and not just one type tp(Ξ±Λ/C). One can check that the definition of f^β1β does not depend on the choice of the tuples Ξ±Λkββ¨qkβ for k<Ξ».
We also leave as an easy exercise to check that if Ξ±Λ contains a (small) model or Ξ±Λ is replaced by cΛ, then GalLfix,1β(p1β) is isomorphic to GalLβ(T), and for Ξ±Λ replaced by cΛ, the above f^β1β can be identified with f^β. The same remark applies to GalKPfix,1β(p1β) and the epimorphism F^1β:EL(SΞ±Λβ(C))βGalKPfix,1β(p1β) (defined before Question 7.19 below).
Proposition 7.18**.**
f^β1β* is a continuous semigroup epimorphism.*
Proof.
The fact that f^β1β is onto is completely standard. It follows from the fact that for every Οβ²βAut(Cβ²) the coset Οβ²AutfLβ(Cβ²) contains some Οβ²β²βAut(Cβ²) which extends an automorphism ΟβAut(C).
Now, let us check continuity. Let CβGalLfix,1β(p1β) be closed. By the definition of the topology, {tp(Οβ²((Ξ±Λkβ)k<Ξ»β)/C):Οβ²β£p1β(Cβ²)β/AutfLfix,1β(p1β(Cβ²))βC} is closed. Hence, D:={β¨tp(Οβ²(Ξ±Λkβ)/C):k<Ξ»β©:Οβ²β£p1β(Cβ²)β/AutfLfix,1β(p1β(Cβ²))βC} is closed in SΞ±Λβ(C)SΞ±Λβ(C). On the other hand, f^β1β1β[C]=EL(SΞ±Λβ(C))β©D. So f^β1β1β[C] is closed.
It remains to check that f^β1β is a homomorphism, which is slightly more delicate in comparison with the proof of the same statement for f^β.
Take Ξ·1β,Ξ·2ββEL(SΞ±Λβ(C)). For each j<Ξ» take a unique kjβ<Ξ» such that Ξ·2β(qjβ)=qkjββ.
By Remark 7.17, there is Ο2β²ββAut(Cβ²) such that
[TABLE]
for all j<Ξ».
Also by Remark 7.17, but applied to the sequence (Ξ²Λβk,jβ)k<Ξ»,jβ€Ξ»β (whose entries satisfy the types qkβ, k<Ξ») defined by
[TABLE]
we can find Ο1β²ββAut(Cβ²) such that
[TABLE]
for all k<Ξ» and jβ€Ξ».
Then (Ο1β²βΟ2β²β)(Ξ±Λjβ)=Ο1β²β(Ξ²kjβ,jβ)β¨Ξ·1β(qkjββ)=Ξ·1β(Ξ·2β(qjβ))=(Ξ·1βΞ·2β)(qjβ)
for all j<Ξ». Hence,
[TABLE]
From the definition of f^β1β, we also get
[TABLE]
By the last two formulas, in order to finish the proof that f^β1β is a homomorphism, it remains to check that
[TABLE]
but this follows from the fact that Ο1β²β(Ξ±Λkβ)=Ο1β²β(Ξ²Λβk,Ξ»β)β¨Ξ·1β(qkβ) for all k<Ξ».
β
Let M1β be a minimal left ideal in EL(SΞ±Λβ(C)) and u1ββM1β an idempotent. Let f1β=f^β1ββ£u1βM1ββ:u1βM1ββGalLfix,1β(p1β). Since u1βM1β=u1βEL(SΞ±Λβ(C))u1β, Proposition 7.18 implies that f1β is a group epimorphism. Let
[TABLE]
A natural counterpart of the Ellis group conjecture in this context says that F1β is an isomorphism. It is false in general (by taking any example where ELβ differs from EKPβ on p1β(C)).
However, one could ask whether it is true assuming that M1β is of bounded size. (Note that then ELβ coincides with EKPβ on p1β(C) by Proposition 7.9 and Corollary 2.10 of [7]).
Question 7.19**.**
Is it true that if M1β is of bounded size, then F1β is an isomorphism?
Actually, most of the above proof of Theorem 0.7 can be easily adjusted to the context of Question 7.19. The only problem that appears is that (as we will see in Example 7.20(6)) it is not the case that each type pβSΞ±Λβ(C) (in a NIP theory) which does not fork over β
is invariant under AutfKP,pfix,1β(C) (the group of automorphisms fixing setwise the EKPβ-class of each realization of p:=tp(Ξ±Λ/β
)). This affects the proof of Lemma 7.15 and the final part of the proof of Lemma 7.16 (namely, we do not have UΒ¬Οβ=UΟcβ). We finish with an example showing that the answer to Question 7.19 is negative. We get even more, namely that the map f1β:u1βM1ββGalLfix,1β(p1β) is not an isomorphism although M1β is finite. In this example, we will compute a minimal left ideal of EL(SΞ±Λβ(C)) and the Ellis group. The notation in this example is not fully compatible with the one used so far (e.g. f1β will denote something else, and the role of p1β will be played by p.)
Example 7.20**.**
Let Qlβ and Qrβ be two disjoint copies of the rationals.
Consider the 2-sorted structure M with the sorts S1β:=QΓQ and S2β:=QlββͺQrβ, the equivalence relation E on S2β with two classes Qlβ and Qrβ,
the order β€S2ββ on S2β which is the standard order on each of the sets Qlβ and Qrβ (and no element of Qlβ is comparable with an element of Qrβ), and the binary relation R on the product S1βΓS2β defined by
[TABLE]
Let T=Th(M) and Cβ»M be a monster model. T is
interpretable in the theory of (Q,β€), hence it is
β΅0β-categorical and has NIP. Also, M is saturated.
Of course, Qlβ as a set is definable over Qlβ treated as an imaginary element (i.e. the βleftβ class of the β
-definable equivalence relation E).
Consider β€1β and β€2β on S1β which are given by the standard orders in the first and in the second coordinate, respectively. Then
[TABLE]
So β€1β and β€2β are both definable over the imaginary element Qlβ, and they are both dense, linear preorders. Let βΌiβ be given by x1ββ€iβx2ββ§x2ββ€iβx1β and β€iβ²β be the induced linear order on S1β/βΌiβ, for i=1,2.
Let f1β:S1ββQlβ and f2β:S1ββQrβ be given by f1β(x)=min{yβQlβ:R(x,y)} and f2β(x)=min{yβQrβ:R(x,y)}.
These functions are definable over the imaginary element Qlβ. Moreover, x1ββ€iβx2ββΊfiβ(x1β)β€iβfiβ(x2β) for i=1,2. Therefore, f1β and f2β induce isomorphisms f1β²β:(S1β/βΌ1β,β€1β²β)β(Qlβ,β€S2ββ) and f2β²β:(S1β/βΌ2β,β€2β²β)β(Qrβ,β€S2ββ). Note that R is definable using β€S2ββ, f1β and f2β:
[TABLE]
Let Ξ±=(0,0)βS1β.
The following statements are true.
- (1)
In each of the sorts S1β and S2β, there is a unique complete type over β
(in the theory T); thus SS1ββ(C)=SΞ±β(C). In fact, Aut(M) acts transitively on S1β and on S2β, and even the group of automorphisms fixing setwise each of the two classes of E acts transitively on S1β. From now on, let p=tp(Ξ±/β
) be the unique type in SS1ββ(β
).
2. (2)
Each automorphism of C which switches the two classes of E also switches the orders β€1β and β€2β. The automorphisms of C which preserve classes of E also preserve β€1β and β€2β.
3. (3)
There is no non-trivial, bounded, invariant equivalence relation on S1β. So ELββ£S1ββ=EKPββ£S1ββ=EShββ£S1ββ=β‘β£S1ββ is the total relation on S1β. Thus, AutfLfix,1β(p(C))=AutfKPfix,1β(p(C))=Aut(p(C)) and GalLfix,1β(p)=GalKPfix,1β(p) is the trivial group.
4. (4)
The quantifier-free {β€1β,β€2β}-type over β
of any finite tuple in S1β generates a complete type over the name of Qlβ in the language of T.
5. (5)
For every Ο΅1β,Ο΅2ββ{0,1} the set of formulas
ZΟ΅1β,Ο΅2ββ:={(xβ€1βa)Ο΅1ββ§(xβ€2βa)Ο΅2β:aβS1β} generates a type
pΟ΅1βΟ΅2βββSS1ββ(C) that does not fork over
β
(here Ο0=Ο
and Ο1=Β¬Ο). These are the only non-forking types
in SS1ββ(C). p00β and p11β are invariant under
Aut(C), and the types p01β and p10β form a single orbit.
6. (6)
The types p01β,p10β are AutfKPβ(C)-invariant (as non-forking types in a NIP theory), but they are not AutfKP,pfix,1β(C)-invariant.
7. (7)
There is Ξ·0ββEL(SΞ±β(C)) such that Im(Ξ·0β)={p00β,p11β,p01β,p10β}.
8. (8)
Each element Ξ·βEL(SΞ±β(C)) acts trivially on p00β,p11β,p01β,p10β or acts trivially on p00β and p11β and switches p01β and p10β, and there exists Ξ· which switches p01β and p10β. Therefore, M1β:=EL(SΞ±β(C))Ξ·0β is a minimal left ideal in EL(SΞ±β(C)), has two elements, and coincides with the Ellis group. Thus, the Ellis group of SΞ±β(C) has two elements, whereas GalKPfix,1β(p) is trivial (by (3)), so the answer to Question 7.19 is negative.
Proof.
(1) Whenever g:S2ββS2β is an {E,β€S2ββ}-automorphism such that glβ:=gβ£Qlββ:QlββQlβ and grβ:=gβ£Qrββ:QrββQrβ, then for f:S1ββS1β given by f(x1β,x2β)=(glβ(x1β),grβ(x2β)) we see that fβͺg is an automorphism of M. Similarly, whenever g:S2ββS2β is an {E,β€S2ββ}-automorphism such that glβ:=gβ£Qlββ:QlββQrβ and grβ:=gβ£Qrββ:QrββQlβ, then for f:S1ββS1β given by f(x1β,x2β)=(grβ(x2β),glβ(x1β)) we see that fβͺg is an automorphism of M. Having this, transitivity of S1β and S2β follows easily from the homogeneity of (Q,β€) and the existence of the obvious {E,β€S2ββ}-automorphism of S2β switching the two E-classes.
(2) is obvious by the descriptions of β€1β and β€2β in terms of Qlβ, Qrβ and R.
(3) First notice that if a0β=(x0β,y0β),a1β=(x1β,y1β)βS1Mβ, then
there are anβ=(xnβ,ynβ)βS1Mβ,nβ₯2 such that the sequence
(anβ) is indiscernible in M. Indeed, it is enough to choose
xnβ,ynβ so that the sequences (xnβ),(ynβ) are either constant
or strictly monotonous. By saturation of M, we see that ELβ is
total on S1β.
(4) It is easy to check it for tuples from S1Mβ (see the proof of (1)), which is enough.
(5)
By (4), for every Ο΅1β,Ο΅2ββ{0,1} and a1β,β¦,anββS1β the formula
[TABLE]
generates a complete type over a1β,β¦,anβ and the name of Qlβ.
So the sets
ZΟ΅1β,Ο΅2ββ generate complete types over C. By
(2), p00β and p11β are invariant and p01β and p10β
are either preserved or switched by any automorphism, so they do not
fork over β
. Now, suppose qβSS1ββ(C)β{p00β,p11β,p01β,p10β}. Then q contains the formula xβ€1βa1ββ§a2ββ€1βx or the formula xβ€2βa1ββ§a2ββ€2βx for some a1β,a2ββS2β (computed in C). But each of these formulas is easily seen to divide over β
.
(6) By (2), any automorphism which switches the E-classes also switches p01β and p10β, but by (3), AutfKP,pfix,1β(C)=Aut(C).
(7)
By (4), for any a1β,β¦,anβ,lβS1β (computed in C), we can choose ΟaΛ,lββAut(C) which preserve the orders β€1β and β€2β and such that ΟaΛ,lβ(aiβ)β€1βl and ΟaΛ,lβ(aiβ)β€2βl for all i=1,β¦,n. Then (ΟaΛ,lβ) is a net with the directed order on the indexes given by: (a1β,β¦,anβ,l)β€(a1β²β,β¦,anβ²β,lβ²) if and only if aΛ is a subsequence of aΛβ² and lβ²β€1βl and lβ²β€2βl. Now, the limit of a convergent subnet will be the desired Ξ·0β.
(8) The first sentence follows from (5). This implies that the image of each element in M1β equals {p00β,p11β,p01β,p10β}, M1β is minimal and has two elements. Hence, any idempotent u1ββM1β fixes p00β,p11β,p01β,p10β, so taking Ξ·βEL(SΞ±β(C)) such that Ξ·(p01β)=p10β, we get that u1βΞ·u1β(p01β)=p10β, and hence u1βM1β has two elements and coincides with M1β.
β
8. Appendix: an example
In this appendix, we give an example showing that in general (vi) does not imply (v) in Proposition 7.9: we construct an NIP structure C and a type p over β
such that p does not fork over β
, however EL(Spβ(C)) has an unbounded minimal ideal. In fact the theory of C will be Ο-categorical.
8.1. Ordered ultrametric spaces
An ordered ultrametric space is a totally ordered set (M,β€) equipped with a distance function d:M2βD, where (D,β΄,0) is a totally ordered set with minimal element [math] such that:
d(x,y)=0βΊx=y;
d(x,y)=d(y,x);
x<y<zβΉd(x,z)=maxβ΄β{d(x,y),d(y,z)}.
As an example of such a space, take (K,v,β€) an ordered valued field with convex valuation ring. Consider the distance d(x,y)=val(xβy) taking value in (Ξ,β₯,β): the value group equipped with the reverse order and β playing the role of [math]. This is an ordered ultrametric space.
Model theoretically, we represent ordered ultrametric spaces as two sorted structures (M,D) in the language L={0,β€,β΄,d}, where [math] is a constant symbol for the minimal element of D, β€ is the order on M, β΄ is the order on D and d:M2βD is the distance function.
Let C be the class of finite ordered ultrametric spaces in the language L (so both sorts are finite).
Proposition 8.1**.**
The class C has the joint embedding and amalgamation properties.
Proof.
Since all structures contain the structure with M empty and D={0}, we only need to show amalgamation. Let A,B,CβC with embedding f:AβB and g:AβC and identify the images of f and g with A. We seek a structure E which completes the square. Let M(A) and D(A) be the two sorts of A and same for B and C. Without loss, D(B) and D(C) both have at least two elements. First amalgamate D(B) and D(C) freely into D(E) (so no element of D(B)βD(A) coincides with an element of D(C)βD(A) and extend the order in an arbitrary way). To amalgamate the M sorts, assume first that M(A) is empty. Define then M(E) as the disjoint union of M(B) and M(C) ordered so that all elements of M(B) are before all elements of M(C). Now if aβM(A)βM(C) and bβM(B)βM(C), set d(a,b)=m, where m=maxβ΄βD(E). This makes E=(M(E),D(E)) into an ordered ultrametric space.
Assume now that M(A) is not empty. Let x<y be two consecutive points in A and set d=d(x,y). Let Bβ² be the points of M(B) strictly between x and y. Define further B0β={zβBβ²:d(x,z)<d} and B1β=Bβ²βB0β. Note that x<B0β<B1β<y according to the order on M(B) and for any zβB0β, zβ²βB1ββͺ{y}, we have d(z,zβ²)=d. Define the same way Cβ²,C0β and C1β, so that x<C0β<C1β<y.
We explain how to amalgamate B0β and C0β. Let bβB0β and cβC0β. If d(x,b)β€d(x,c), set b<c and d(b,c)=d(x,c). If d(x,c)<d(x,b), then set c<b and d(c,b)=d(x,b). We amalgamate B1β and C1β similarly by considering d(β
,y): if d(b,y)β€d(c,y), set c<b and d(c,b)=d(c,y) and symmetrically if d(c,y)<d(b,y). Finally, if bβB0ββͺC0β and cβB1ββͺC1β, then set b<c and d(b,c)=d.
Now, if bβM(B) and cβM(C) have a point xβM(A) strictly between them, say b<x<c, set b<c and d(b,c)=max{d(b,x),d(x,c)}. One can check that this does not depend on the choice of x. Finally, if b,c and both greater than the greatest point x of M(A), then one amalgamates them with the same rules as for B0β and C0β above (and symmetrically if b,c and smaller than the minimum of M(A)). It is now straightforward to check that this does define an ordered ultrametric space C as required.
β
8.2. Trees and betweenness relation
By a meet-tree, we mean a partially ordered set (T,β€) such that, for every element aβT, the set {xβT:xβ€a} is linearly ordered by β€ and every two-element subset {a,b} of T, has a greatest lower bound aβ§b. This theory has a model companion: the theory Tdβ of dense meet-trees which is Ο-categorical and admits elimination of quantifiers in the language {β€,β§}.
Given a meet-tree (T,β€,β§), we define the tree-betweenness relation B(x,y,z) which holds for (a,b,c)βT3 if and only if b is in the path linking a to c, that is:
[TABLE]
By an interval of (T,B), we mean a set of the form [a,b]:={xβT:B(a,x,b)} for some a,bβT. Such an interval is equipped with a natural definable linear ordering where c is less than d if B(a,c,d) (if we represent the interval as [b,a] instead of [a,b], we obtain the opposite ordering, with b as the first point).
Given three distinct points a,b,cβT, the intersection [a,b]β©[b,c]β©[a,c] has a unique point;
call this point the meet of (a,b,c) and denote it by β(a,b,c). If some two of a,b,c are equal, then define β(a,b,c) as being equal to their common value. Fix some aβT. Given b,cβT, write bEaβc if aβ/[b,c]. Then Eaβ is an equivalence relation on Tβ{a}. The equivalence classes of Eaβ are called cones at a.
For the purposes of this example, a (tree) betweenness structure is the reduct to (B,β) of a meet-tree (T,β€,β§). If (T,β€,β§)β¨Tdβ, then the reduct (T,B,β) to the betweenness relation admits elimination of quantifiers in the language (B,β) and its automorphism group acts transitively on pairs of distinct elements.
We note some basic properties of trees and betweenness relations (the reader is encouraged to make drawings to follow the statements).
If (T,B,β) is a tree betweenness structure and aβT is any point, then one can define a meet-tree structure with a as minimal element by setting bβ€cβΊB(a,b,c). Taking the betweenness relation associated to this tree yields back B.
Let (T,B,β) be a tree betweenness structure and let T0ββT be a finite set. Then {aβT:a=β(c,d,e) for some c,d,eβT0β} is a substructure of T. In particular, the substructure generated by T0β has size at most β£T0ββ£3.
Let T0ββT be a finite substructure and aβT a point. The possibilities for qftp(a/T0β) are as follows:
- (1)
T0ββͺ{a} is a substructure. This splits into three cases:
- (a)
aβT0β;
2. (b)
a lies in an interval of T0β: there are b,cβT0β such that aβ[b,c]. The knowledge of a minimal such interval completely determines qftp(a/T0β).
3. (c)
a does not lie in an interval of T0β, then there is a unique bβT0β such that [a,b] contains no other point of T0β and for every cβT0β, bβ[a,c] (b is the point in T0β closest to a). The knowledge of b determines qftp(a/T0β).
2. (2)
T0ββͺ{a} is not a substructure. Then there are b,cβT0β such that aββ:=β(a,b,c) is a new element. Then aβββ[b,c], so its type belongs to the case (1b) above. Also qftp(a/T0βaββ) belongs to case (1c) above, with aββ being the closest element to a in the tree T0ββͺ{aββ}. In particular aβββ[b,a] for every bβT0β and T0ββͺ{a,aββ} is a substructure.
In this case, qftp(a/T0β) is determined by qftp(aββ/T0β).
Note that it follows from this analysis that there are at most β£T0ββ£+β£T0ββ£2+β£T0ββ£+β£T0ββ£2=O(β£T0ββ£2) quantifier-free one types over T0β.
8.3. The example
We work in a language L with two sorts M and D. In addition to those sorts, we have in our language a ternary relation B(x,y,z) and a ternary function symbol β(x,y,z), all on the main sort M, a function d:M2βD, a binary relation β΄ on D and a constant [math] of sort D.
Let C0β be the class of finite L-structures (M,D;B,β,d,β΄,0), where:
B is a tree-betweenness relation on M and β is the meet in that structure;
β΄ is a linear order on D with minimal element [math];
d:M2βD is such that for any two elements x,yβM, the interval [x,y] equipped with d is an ordered ultrametric space as described above (this condition is invariant under reversing the ordering on [x,y]).
Note that the third bullet implies that d is an ultrametric distance on M: for any x,y,zβM, d(x,y)β€max{d(y,z),d(x,z)}.
Lemma 8.2**.**
The class C0β is a FraΓ―ssΓ© class.
Proof.
Tree-betweenness relations are closed under substructures, as follows for instance from the first bullet above. Hence C0β is a universal class.
As previously, we can skip joint embedding and only check amalgamation. Assume E,F,GβC0β are substructures with EβF and EβG. Without loss, M(F) and M(G) are not empty. We first amalgamate the sorts D freely as in the case of ordered ultrametric spaces. We then amalgamate the main sorts. If M(E) is empty, then arbitrarily fix point aβM(F) and aβ²βM(G) and identify them, adding a point to M(E) that maps to both. So we can assume that M(E) is non-empty. Next, it is enough to consider the case where M(F) is generated over M(E) by a single element a and similarly M(G) is generated over M(E) by a single element b (then induct on the size of M(F)βM(E) and M(G)βM(E)).
We will use the list of (quantifier-free) types in betweenness relations presented above. If either a or b is of class (2), define aββ and bββ as done there. Otherwise, set aββ=a (resp. bββ=b). Assume that the types of both aββ and bββ over M(E) are of class (1b) in that list and both land in the same minimal interval [x,y] of M(E). Then that interval, with distance d is an ordered ultrametric space, hence we can amalgamate those points as explained above. Assume next that both types of aββ and bββ are of class (1c) and have the same closest point x in M(E). Amalgamate aββ and bββ over M(E) so that Β¬(aββExβbββ) holds and set d(aββ,bββ)=max{d(aββ,x),d(bββ,x)}. In all other cases, there is a unique way to extend the tree structure to M(E)βͺ{aββ,bββ}. Having done this, there is xβM(E) in the interval [aββ,bββ] and we can (and must) set d(aββ,bββ)=maxβ΄β{d(aββ,x),d(bββ,x)}.
Let Eβ² be the resulting structure. Then M(F) and M(G) are generated over Eβ² by a and b respectively and the types of a and b over M(Eβ²) are of class (1). We can therefore repeat the procedure to amalgamate them. This finishes the proof.
β
Let U be the FraΓ―ssΓ© limit of C0β and C a monster model of it. It is an Ο-categorical structure which admits elimination of quantifiers in the language L.
Lemma 8.3**.**
The structure U is NIP.
Proof.
Recall that a theory is NIP if and only if for every formula Ο(x,yΛβ), there is k such that the number of Ο-types over any finite set A is bounded by β£Aβ£k. We will in fact check that there are polynomially many 1-types over finite sets. So let AβC be a finite set and cβC. As the subtree generated by M(A) has size polynomial in A, we can assume that M(A) is a subtree. Then we can close A under the distance function and hence assume that A is a substructure.
If c is of sort D, then by quantifier elimination, there are at most 2β£D(A)β£+1 possibilities for tp(c/A).
Next assume that c is in the main sort M. The subtree generated by c over M(A) has size either β£M(A)β£+1 or β£M(A)β£+2. If the type of c over M(A) is of class (2) above, then tp(c/M(A)) is determined by tp(cββ/M(A)) and tp(c/M(A)cββ). Each of those types is of class (1) and we can therefore reduce to this case and assume that M(A)βͺ{c} is already a subtree. Let Dβ² be the set of distances of pairs of elements in M(A)βͺ{c}. Then Dβ²βD contains at most one element. By the previous paragraph, there are polynomially many possibilities for the type of that element over D. Adding that element to A, we can assume that for all a,bβM(A)βͺ{c}, d(a,b)βD(A).
We know from the analysis of types above that there are only polynomially many 1-types over A in the reduct to the tree. As for the distance: take aβM(A) for which d(a,c) is minimal. Then the data of a and d(a,c) determines all distances d(b,c), bβM(A), using the ultrametric identity. Indeed, if d(a,b)βΉd(a,c), then d(b,c)=d(a,b) and otherwise d(b,c)=d(a,c).
This discussion shows that there are polynomially many possibilities for tp(c/A) and we conclude that U is NIP.
β
Note that by quantifier elimination (and Ο-categoricity), the automorphism group of U acts transitively on D(U)β{0} and on pairs of distinct points of M(U). It follows that for each aβM(U), the stabilizer of a acts transitively on the cones at a. Let B be the imaginary sort of d-balls of the form {xβM:d(x,a)<r} for some aβM and rβDβ{0}. Let β be the definable binary relation in MΓB representing membership of an element in a ball and let r:BβD give the radius of a ball (definable as r(b)=maxβ΄β{d(a,b):a,bβb}). There is a unique type p(x) over β
of an element of B (since the automorphism group of U acts transitively on pairs (a,r)βM(U)ΓD(U)).
By quantifier elimination, there is a unique type r0β(t) in S(C) of sort D containing tβΉd for all dβD(C). Pick any aβM(C) and let p0β(x) be the type over C of a ball b containing a with radius r(b)β¨r0β. This does not depend on the choice of a since such a ball contains all points in M(C). The type p0β is thus invariant under Aut(C).
Let Q be the set of global extensions q(x) of p(x) which satisfy r(x)βd for some dβD(C)β{0}. Pick a type qβQ and dβD(C)β{0} such that qβ’r(x)βd. Let bβ¨q. Assume first that the ball coded by b contains a point aβM(C). One can find unboundedly many points in C at distance at least d from each other. For each ΟβAut(C), the ball coded by Ο(q) can contain at most one of those points. Since the automorphism group of C acts transitively on the main sort, q has unbounded Aut(C)-orbit. If now b does not contain a point in M(C), pick some aβM(C). Then all elements in the ball coded by b lie in the same cone at a and therefore b determines a cone at a. If this cone contains an M(C)-point, then q has unbounded Aut(C)-orbit, as there are unboundedly many cones at a and Aut(C/a) acts transitively on them. Assume that this is not the case and take aβ²βM(C), aβ²ξ =a. Then for any point d in the ball b (in some larger monster model), a lies in the interval [aβ²,d]. It follows that the ball b is included in the cone at aβ² defined by a and we are reduce to the previous case by changing a to aβ². Hence, in all cases, q forks over β
. Let d0ββD(C)β{0}. The formula r(x)βd0β is weakly invariant as there is qβQ with q(x)β’r(x)βd for all dβD(C)β{0}, but that formula forks over β
. By Corollary 7.4, the minimal ideal of EL(Spβ(C)) is not bounded. However, p does not fork over β
since p0β is an invariant extension of it.
Acknowledgments
We would like to thank to the referee for very careful reading and all the comments and suggestions which helped us to improve presentation.
